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1 Chapter 1 1. THINK In this problem we’re given the radius of Earth, and asked to compute its circumference, surface area and volume. EXPRESS Assuming Earth to be a sphere of radius 6 3 36.37 10 m 10 km m 6.37 10 km,ER the corresponding circumference, surface area and volume are: 2 342 , 4 , 3 E E EC R A R V R . The geometric formulas are given in Appendix E. ANALYZE (a) Using the formulas given above, we find the circumference to be 3 42 2 (6.37 10 km) 4.00 10 km.EC R (b) Similarly, the surface area of Earth is 2 2 3 8 24 4 6.37 10 km 5.10 10 kmEA R , (c) and its volume is 3 3 3 12 34 4 6.37 10 km 1.08 10 km . 3 3 EV R LEARN From the formulas given, we see that EC R , 2 EA R , and 3 EV R . The ratios of volume to surface area, and surface area to circumference are / / 3EV A R and / 2 EA C R . 2. The conversion factors are: 1 gry 1/10 line , 1 line 1/12 inch and 1 point = 1/72 inch. The factors imply that 1 gry = (1/10)(1/12)(72 points) = 0.60 point. Thus, 1 gry 2 = (0.60 point) 2 = 0.36 point 2 , which means that 2 20.50 gry = 0.18 point . 3. The metric prefixes (micro, pico, nano, …) are given for ready reference on the inside front cover of the textbook (see also Table 1–2). CHAPTER 1 2 (a) Since 1 km = 1 103 m and 1 m = 1 106 m, 3 3 6 91km 10 m 10 m 10 m m 10 m. The given measurement is 1.0 km (two significant figures), which implies our result should be written as 1.0 109 m. (b) We calculate the number of microns in 1 centimeter. Since 1 cm = 10 2 m, 2 2 6 41cm = 10 m = 10 m 10 m m 10 m. We conclude that the fraction of one centimeter equal to 1.0 m is 1.0 104. (c) Since 1 yd = (3 ft)(0.3048 m/ft) = 0.9144 m, 6 51.0yd = 0.91m 10 m m 9.1 10 m. 4. (a) Using the conversion factors 1 inch = 2.54 cm exactly and 6 picas = 1 inch, we obtain 1 inch 6 picas 0.80 cm = 0.80 cm 1.9 picas. 2.54 cm 1 inch (b) With 12 points = 1 pica, we have 1 inch 6 picas 12 points 0.80 cm = 0.80 cm 23 points. 2.54 cm 1 inch 1 pica 5. THINK This problem deals with conversion of furlongs to rods and chains, all of which are units for distance. EXPRESS Given that 1 furlong 201.168 m, 1rod 5.0292 m and 1chain 20.117 m , the relevant conversion factors are 1 rod 1.0 furlong 201.168 m (201.168 m) 40 rods, 5.0292 m and 1 chain 1.0 furlong 201.168 m (201.168 m) 10 chains 20.117 m . Note the cancellation of m (meters), the unwanted unit. ANALYZE Using the above conversion factors, we find (a) the distance d in rods to be 40 rods 4.0 furlongs 4.0 furlongs 160 rods, 1 furlong d 3 (b) and in chains to be 10 chains 4.0 furlongs 4.0 furlongs 40 chains. 1 furlong d LEARN Since 4 furlongs is about 800 m, this distance is approximately equal to 160 rods ( 1 rod 5 m ) and 40 chains ( 1 chain 20 m ). So our results make sense. 6. We make use of Table 1-6. (a) We look at the first (“cahiz”) column: 1 fanega is equivalent to what amount of cahiz? We note from the already completed part of the table that 1 cahiz equals a dozen fanega. Thus, 1 fanega = 1 12 cahiz, or 8.33 102 cahiz. Similarly, “1 cahiz = 48 cuartilla” (in the already completed part) implies that 1 cuartilla = 1 48 cahiz, or 2.08 102 cahiz. Continuing in this way, the remaining entries in the first column are 6.94 103 and 33.47 10 . (b) In the second (“fanega”) column, we find 0.250, 8.33 102, and 4.17 102 for the last three entries. (c) In the third (“cuartilla”) column, we obtain 0.333 and 0.167 for the last two entries. (d) Finally, in the fourth (“almude”) column, we get 1 2 = 0.500 for the last entry. (e) Since the conversion table indicates that 1 almude is equivalent to 2 medios, our amount of 7.00 almudes must be equal to 14.0 medios. (f) Using the value (1 almude = 6.94 103 cahiz) found in part (a), we conclude that 7.00 almudes is equivalent to 4.86 102 cahiz. (g) Since each decimeter is 0.1 meter, then 55.501 cubic decimeters is equal to 0.055501 m 3 or 55501 cm 3 . Thus, 7.00 almudes = 7.00 12 fanega = 7.00 12 (55501 cm 3 ) = 3.24 104 cm3. 7. We use the conversion factors found in Appendix D. 2 31 acre ft = (43,560 ft ) ft = 43,560 ft Since 2 in. = (1/6) ft, the volume of water that fell during the storm is 2 2 2 7 3(26 km )(1/6 ft) (26 km )(3281ft/km) (1/6 ft ) 4.66 10 ft .V Thus, V 4 66 10 4 3560 10 11 10 7 4 3. . . ft ft acre ft acre ft. 3 3 CHAPTER 1 4 8. From Fig. 1-4, we see that 212 S is equivalent to 258 W and 212 – 32 = 180 S is equivalent to 216 – 60 = 156 Z. The information allows us to convert S to W or Z. (a) In units of W, we have 258 W 50.0 S 50.0 S 60.8 W 212 S (b) In units of Z, we have 156 Z 50.0 S 50.0 S 43.3 Z 180 S 9. The volume of ice is given by the product of the semicircular surface area and the thickness. The area of the semicircle is A = r2/2, where r is the radius. Therefore, the volume is 2 2 V r z where z is the ice thickness. Since there are 10 3 m in 1 km and 10 2 cm in 1 m, we have 3 2 510 m 10 cm2000km 2000 10 cm. 1km 1m r In these units, the thickness becomes 2 210 cm3000m 3000m 3000 10 cm 1m z which yields 2 5 2 22 32000 10 cm 3000 10 cm 1.9 10 cm . 2 V 10. Since a change of longitude equal to 360 corresponds to a 24 hour change, then one expects to change longitude by 360 / 24 15 before resetting one's watch by 1.0 h. 11. (a) Presuming that a French decimal day is equivalent to a regular day, then the ratio of weeks is simply 10/7 or (to 3 significant figures) 1.43. (b) In a regular day, there are 86400 seconds, but in the French system described in the problem, there would be 10 5 seconds. The ratio is therefore 0.864. 12. A day is equivalent to 86400 seconds and a meter is equivalent to a million micrometers, so 5 37 10 14 86400 31 6. . . m m m day s day m s b gc h b gb g 13. The time on any of these clocks is a straight-line function of that on another, with slopes 1 and y-intercepts 0. From the data in the figure we deduce 2 594 33 662 , . 7 7 40 5 C B B At t t t These are used in obtaining the following results. (a) We find 33 495 s 40 B B A At t t t when t'A tA = 600 s. (b) We obtain t t t tC C B B 2 7 2 7 495 141b g b g s. (c) Clock B readstB = (33/40)(400) (662/5) 198 s when clock A reads tA = 400 s. (d) From tC = 15 = (2/7)tB + (594/7), we get tB 245 s. 14. The metric prefixes (micro (), pico, nano, …) are given for ready reference on the inside front cover of the textbook (also Table 1–2). (a) 6 100 y 365 day 24 h 60 min 1 century 10 century 52.6 min. 1 century 1 y 1 day 1 h (b) The percent difference is therefore 52.6 min 50 min 4.9%. 52.6 min 15. A week is 7 days, each of which has 24 hours, and an hour is equivalent to 3600 seconds. Thus, two weeks (a fortnight) is 1209600 s. By definition of the micro prefix, this is roughly 1.21 1012 s. 16. We denote the pulsar rotation rate f (for frequency). 3 1 rotation 1.55780644887275 10 s f CHAPTER 1 6 (a) Multiplying f by the time-interval t = 7.00 days (which is equivalent to 604800 s, if we ignore significant figure considerations for a moment), we obtain the number of rotations: 3 1 rotation 604800 s 388238218.4 1.55780644887275 10 s N which should now be rounded to 3.88 108 rotations since the time-interval was specified in the problem to three significant figures. (b) We note that the problem specifies the exact number of pulsar revolutions (one million). In this case, our unknown is t, and an equation similar to the one we set up in part (a) takes the form N = ft, or 6 3 1 rotation 1 10 1.55780644887275 10 s t which yields the result t = 1557.80644887275 s (though students who do this calculation on their calculator might not obtain those last several digits). (c) Careful reading of the problem shows that the time-uncertainty per revolution is 173 10 s . We therefore expect that as a result of one million revolutions, the uncertainty should be 17 6 11 ( 3 10 )(1 10 )= 3 10 s . 17. THINK In this problem we are asked to rank 5 clocks, based on their performance as timekeepers. EXPRESS We first note that none of the clocks advance by exactly 24 h in a 24-h period but this is not the most important criterion for judging their quality for measuring time intervals. What is important here is that the clock advance by the same (or nearly the same) amount in each 24-h period. The clock reading can then easily be adjusted to give the correct interval. ANALYZE The chart below gives the corrections (in seconds) that must be applied to the reading on each clock for each 24-h period. The entries were determined by subtracting the clock reading at the end of the interval from the clock reading at the beginning. Clocks C and D are both good timekeepers in the sense that each is consistent in its daily drift (relative to WWF time); thus, C and D are easily made “perfect” with simple and predictable corrections. The correction for clock C is less than the correction for clock D, so we judge clock C to be the best and clock D to be the next best. The correction that must be applied to clock A is in the range from 15 s to 17s. For clock B it is the range from 5 s to +10 s, for clock E it is in the range from 70 s to 2 s. After C and D, A has 7 the smallest range of correction, B has the next smallest range, and E has the greatest range. From best to worst, the ranking of the clocks is C, D, A, B, E. CLOCK Sun. Mon. Tues. Wed. Thurs. Fri. -Mon. -Tues. -Wed. -Thurs. -Fri. -Sat. A 16 16 15 17 15 15 B 3 +5 10 +5 +6 7 C 58 58 58 58 58 58 D +67 +67 +67 +67 +67 +67 E +70 +55 +2 +20 +10 +10 LEARN Of the five clocks, the readings in clocks A, B and E jump around from one 24- h period to another, making it difficult to correct them. 18. The last day of the 20 centuries is longer than the first day by 20 century 0.001 s century 0.02 s. The average day during the 20 centuries is (0 + 0.02)/2 = 0.01 s longer than the first day. Since the increase occurs uniformly, the cumulative effect T is average increase in length of a day number of days 0.01 s 365.25 day 2000 y day y 7305 s T or roughly two hours. 19. When the Sun first disappears while lying down, your line of sight to the top of the Sun is tangent to the Earth’s surface at point A shown in the figure. As you stand, elevating your eyes by a height h, the line of sight to the Sun is tangent to the Earth’s surface at point B. Let d be the distance from point B to your eyes. From the Pythagorean theorem, we have 2 2 2 2 2( ) 2d r r h r rh h CHAPTER 1 8 or 2 22 ,d rh h where r is the radius of the Earth. Since r h , the second term can be dropped, leading to 2 2d rh . Now the angle between the two radii to the two tangent points A and B is , which is also the angle through which the Sun moves about Earth during the time interval t = 11.1 s. The value of can be obtained by using 360 24 h t . This yields (360 )(11.1 s) 0.04625 . (24 h)(60 min/h)(60 s/min) Using tand r , we have 2 2 2tan 2d r rh , or 2 2 tan h r Using the above value for and h = 1.7 m, we have 65.2 10 m.r 20. (a) We find the volume in cubic centimeters 3 3 5 3231 in 2.54 cm193 gal = 193 gal 7.31 10 cm 1gal 1in and subtract this from 1 106 cm3 to obtain 2.69 105 cm3. The conversion gal in3 is given in Appendix D (immediately below the table of Volume conversions). (b) The volume found in part (a) is converted (by dividing by (100 cm/m) 3 ) to 0.731 m 3 , which corresponds to a mass of 1000 kg m 0.731 m = 731 kg3 2c h c h using the density given in the problem statement. At a rate of 0.0018 kg/min, this can be filled in 5731kg 4.06 10 min = 0.77 y 0.0018 kg min after dividing by the number of minutes in a year (365 days)(24 h/day) (60 min/h). 21. If ME is the mass of Earth, m is the average mass of an atom in Earth, and N is the number of atoms, then ME = Nm or N = ME/m. We convert mass m to kilograms using Appendix D (1 u = 1.661 1027 kg). Thus, 9 N M m E 598 10 40 1661 10 9 0 10 24 27 49. . . . kg u kg ub g c h 22. The density of gold is 3 3 19.32 g 19.32 g/cm . 1 cm m V (a) We take the volume of the leaf to be its area A multiplied by its thickness z. With density = 19.32 g/cm3 and mass m = 27.63 g, the volume of the leaf is found to be V m 1430. .cm3 We convert the volume to SI units: 3 3 6 31 m1.430 cm 1.430 10 m . 100 cm V Since V = Az with z = 1 10-6 m (metric prefixes can be found in Table 1–2), we obtain A 1430 10 1 10 1430 6 6 . . . m m m 3 2 (b) The volume of a cylinder of length is V A where the cross-section area is that of a circle: A = r2. Therefore, with r =2.500 106 m and V = 1.430 106 m3, we obtain 4 2 7.284 10 m 72.84 km. V r 23. THINK This problem consists of two parts: in the first part, we are asked to find the mass of water, given its volume and density; the second part deals with the mass flow rate of water, which is expressed as kg/s in SI units. EXPRESS From the definition of density: /m V , we see that mass can be calculated as m V , the product of the volume of water and its density. With 1 g = 1 103 kg and 1 cm 3 = (1 102m)3 = 1 106m3, the density of water in SI units (kg/m3) is 3 3 3 3 3 3 6 3 1 g 10 kg cm 1 g/cm 1 10 kg m . cm g 10 m To obtain the flow rate, we simply divide the total mass of the water by the time taken to drain it. ANALYZE (a) Using m V , the mass of a cubic meter of water is CHAPTER 1 10 3 3 3(1 10 kg/m )(1m ) 1000 kg.m V (b) The total mass of water in the container is 3 3 3 6(1 10 kg m )(5700 m ) 5.70 10 kgM V , and the time elapsed is t = (10 h)(3600 s/h) = 3.6 104 s. Thus, the mass flow rate R is 6 4 5.70 10 kg 158 kg s. 3.6 10 s M R t LEARN In terms of volume, the drain rate can be expressed as 3 3 4 5700 m 0.158 m /s 42 gal/s. 3.6 10 s V R t The greater the flow rate, the less time required to drain a given amount of water. 24. The metric prefixes (micro (), pico, nano, …) are given for ready reference on the inside front cover of the textbook (see also Table 1–2). The surface area A of each grain of sand of radius r = 50 m = 50 106 m is given by A = 4(50 106)2 = 3.14 108 m 2 (Appendix E contains a variety of geometry formulas). We introduce the notion of density, /m V , so that the mass can be found from m = V, where = 2600 kg/m3. Thus, using V = 4r3/3, the mass of each grain is 3 63 9 3 4 50 10 m4 kg 2600 1.36 10 kg. 3 m 3 r m V We observe that (because a cube has six equal faces) the indicated surface area is 6 m 2 . The number of spheres (the grains of sand) N that have a total surface area of 6 m 2 is given by 2 8 8 2 6 m 1.91 10 . 3.14 10 m N Therefore, the total mass M is 8 91.91 10 1.36 10 kg 0.260 kg.M Nm 25. The volume of the section is (2500 m)(800 m)(2.0 m) = 4.0 106 m3. Letting “d” stand for the thickness of the mud after it has (uniformly) distributed in the valley, then its volume there would be (400 m)(400 m)d. Requiring these two volumes to be equal, we can solve for d. Thus, d = 25 m. The volume of a small part of the mud over a patch of area of 4.0 m 2 is (4.0)d = 100 m 3 . Since each cubic meter corresponds to a mass of 11 1900 kg (stated in the problem), then the mass of that small part of the mud is 51.9 10 kg . 26. (a) The volume of the cloud is (3000 m)(1000 m)2 = 9.4 109 m3. Since each cubic meter of the cloud contains from 50 106 to 500 106 water drops, then we conclude that the entire cloud contains from 4.7 1018 to 4.7 1019 drops. Since the volume of each drop is 4 3 (10 106 m)3 = 4.2 1015 m3, then the total volume of water in a cloud is from 32 10 to 42 10 m 3 . (b) Using the fact that 3 3 3 31 L 1 10 cm 1 10 m , the amount of water estimated in part (a) would fill from 62 10 to 72 10 bottles. (c) At 1000 kg for every cubic meter, the mass of water is from 62 10 to 72 10 kg. The coincidence in numbers between the results of parts (b) and (c) of this problem is due to the fact that each liter has a mass of one kilogram when water is at its normal density (under standard conditions). 27. We introduce the notion of density, /m V , and convert to SI units: 1000 g = 1 kg, and 100 cm = 1 m. (a) The density of a sample of iron is 3 3 31 kg 100 cm7.87 g cm 7870 kg/m . 1000 g 1 m If we ignore the empty spaces between the close-packed spheres, then the density of an individual iron atom will be the same as the density of any iron sample. That is, if M is the mass and V is the volume of an atom, then 26 29 3 3 3 9.27 10 kg 1.18 10 m . 7.87 10 kg m M V (b) We set V = 4R3/3, where R is the radius of an atom (Appendix E contains several geometry formulas). Solving for R, we find 1 3 29 31 3 10 3 1.18 10 m3 1.41 10 m. 4 4 V R The center-to-center distance between atoms is twice the radius, or 2.82 1010 m. CHAPTER 1 12 28. If we estimate the “typical” large domestic cat mass as 10 kg, and the “typical” atom (in the cat) as 10 u 2 1026 kg, then there are roughly (10 kg)/( 2 1026 kg) 5 10 26 atoms. This is close to being a factor of a thousand greater than Avogadro’s number. Thus this is roughly a kilomole of atoms. 29. The mass in kilograms is 28 9 100 16 10 10 0 3779 . . piculs gin 1picul tahil 1gin chee 1tahil hoon 1 chee g 1hoon b g FHG I KJ F HG I KJ F HG I KJ F HG I KJ F HG I KJ which yields 1.747 106 g or roughly 1.75 103 kg. 30. To solve the problem, we note that the first derivative of the function with respect to time gives the rate. Setting the rate to zero gives the time at which an extreme value of the variable mass occurs; here that extreme value is a maximum. (a) Differentiating 0.8( ) 5.00 3.00 20.00m t t t with respect to t gives 0.24.00 3.00. dm t dt The water mass is the greatest when / 0,dm dt or at 1/0.2(4.00 /3.00) 4.21s.t (b) At 4.21s,t the water mass is 0.8( 4.21s) 5.00(4.21) 3.00(4.21) 20.00 23.2 g.m t (c) The rate of mass change at 2.00 st is 0.2 2.00 s 2 g 1 kg 60 s 4.00(2.00) 3.00 g/s 0.48 g/s 0.48 s 1000 g 1 min 2.89 10 kg/min. t dm dt (d) Similarly, the rate of mass change at 5.00 st is 0.2 2.00 s 3 g 1 kg 60 s 4.00(5.00) 3.00 g/s 0.101g/s 0.101 s 1000 g 1 min 6.05 10 kg/min. t dm dt 31. The mass density of the candy is 13 4 3 4 3 3 0.0200 g 4.00 10 g/mm 4.00 10 kg/cm . 50.0 mm m V If we neglect the volume of the empty spaces between the candies, then the total mass of the candies in the container when filled to height h is ,M Ah where 2(14.0 cm)(17.0 cm) 238 cmA is the base area of the container that remains unchanged. Thus, the rate of mass change is given by 4 3 2( ) (4.00 10 kg/cm )(238 cm )(0.250 cm/s) 0.0238 kg/s 1.43 kg/min. dM d Ah dh A dt dt dt 32. The total volume V of the real house is that of a triangular prism (of height h = 3.0 m and base area A = 20 12 = 240 m2) in addition to a rectangularbox (height h´ = 6.0 m and same base). Therefore, 31 1800 m . 2 2 h V hA h A h A (a) Each dimension is reduced by a factor of 1/12, and we find Vdoll 3 3m m F HG I KJ 1800 1 12 10 3 c h . . (b) In this case, each dimension (relative to the real house) is reduced by a factor of 1/144. Therefore, Vminiature 3m 6.0 10 m F HG I KJ 1800 1 144 3 4 3c h . 33. THINK In this problem we are asked to differentiate between three types of tons: displacement ton, freight ton and register ton, all of which are units of volume. EXPRESS The three different tons are defined in terms of barrel bulk, with 31 barrel bulk 0.1415 m 4.0155 U.S. bushels (using 31 m 28.378 U.S. bushels ). Thus, in terms of U.S. bushels, we have 4.0155 U.S. bushels 1 displacement ton (7 barrels bulk) 28.108 U.S. bushels 1 barrel bulk 4.0155 U.S. bushels 1 freight ton (8 barrels bulk) 32.124 U.S. bushels 1 barrel bulk 4.0155 U.S. bushels 1 register ton (20 barrels bulk) 80.31 U.S. bushels 1 barrel bulk CHAPTER 1 14 ANALYZE (a) The difference between 73 “freight” tons and 73 “displacement” tons is 73(freight tons displacement tons) 73(32.124 U.S. bushels 28.108 U.S. bushels) 293.168 U.S. bushels 293 U.S. bushels V (b) Similarly, the difference between 73 “register” tons and 73 “displacement” tons is 3 73(register tons displacement tons) 73(80.31 U.S. bushels 28.108 U.S. bushels) 3810.746 U.S. bushels 3.81 10 U.S. bushels V LEARN With 1 register ton 1 freight ton 1displacement ton, we expect the difference found in (b) to be greater than that in (a). This is indeed the case. 34. The customer expects a volume V1 = 20 7056 in 3 and receives V2 = 20 5826 in. 3 , the difference being 3 1 2 24600 in.V V V , or 3 3 3 2.54cm 1L 24600 in. 403L 1 inch 1000 cm V where Appendix D has been used. 35. The first two conversions are easy enough that a formal conversion is not especially called for, but in the interest of practice makes perfect we go ahead and proceed formally: (a) 2 peck 11 tuffets = 11 tuffets 22 pecks 1 tuffet . (b) 0.50 Imperial bushel 11 tuffets = 11 tuffets 5.5 Imperial bushels 1 tuffet . (c) 36.3687 L 11 tuffets = 5.5 Imperial bushel 200 L 1 Imperial bushel . 36. Table 7 can be completed as follows: (a) It should be clear that the first column (under “wey”) is the reciprocal of the first row – so that 9 10 = 0.900, 3 40 = 7.50 102, and so forth. Thus, 1 pottle = 1.56 103 wey and 1 gill = 8.32 106 wey are the last two entries in the first column. (b) In the second column (under “chaldron”), clearly we have 1 chaldron = 1 chaldron (that is, the entries along the “diagonal” in the table must be 1’s). To find out how many 15 chaldron are equal to one bag, we note that 1 wey = 10/9 chaldron = 40/3 bag so that 1 12 chaldron = 1 bag. Thus, the next entry in that second column is 1 12 = 8.33 102. Similarly, 1 pottle = 1.74 103 chaldron and 1 gill = 9.24 106 chaldron. (c) In the third column (under “bag”), we have 1 chaldron = 12.0 bag, 1 bag = 1 bag, 1 pottle = 2.08 102 bag, and 1 gill = 1.11 104 bag. (d) In the fourth column (under “pottle”), we find 1 chaldron = 576 pottle, 1 bag = 48 pottle, 1 pottle = 1 pottle, and 1 gill = 5.32 103 pottle. (e) In the last column (under “gill”), we obtain 1 chaldron = 1.08 105 gill, 1 bag = 9.02 103 gill, 1 pottle = 188 gill, and, of course, 1 gill = 1 gill. (f) Using the information from part (c), 1.5 chaldron = (1.5)(12.0) = 18.0 bag. And since each bag is 0.1091 m 3 we conclude 1.5 chaldron = (18.0)(0.1091) = 1.96 m 3 . 37. The volume of one unit is 1 cm 3 = 1 106 m3, so the volume of a mole of them is 6.02 1023 cm3 = 6.02 1017 m3. The cube root of this number gives the edge length: 5 38.4 10 m . This is equivalent to roughly 8 102 km. 38. (a) Using the fact that the area A of a rectangle is (width) length), we find total 2 2 3.00acre 25.0perch 4.00perch 40 perch 4 perch 3.00 acre 100perch 1acre 580 perch . A We multiply this by the perch 2 rood conversion factor (1 rood/40 perch2) to obtain the answer: Atotal = 14.5 roods. (b) We convert our intermediate result in part (a): 2 2 5 2 total 16.5ft 580 perch 1.58 10 ft . 1perch A Now, we use the feet meters conversion given in Appendix D to obtain 2 5 2 4 2 total 1m 1.58 10 ft 1.47 10 m . 3.281ft A CHAPTER 1 16 39. THINK This problem compares the U.K. gallon with U.S. gallon, two non-SI units for volume. The interpretation of the type of gallons, whether U.K. or U.S., affects the amount of gasoline one calculates for traveling a given distance. EXPRESS If the fuel consumption rate is R (in miles/gallon), then the amount of gasoline (in gallons) needed for a trip of distance d (in miles) would be (miles) (gallon) (miles/gallon) d V R Since the car was manufactured in U.K., the fuel consumption rate is calibrated based on U.K. gallon, and the correct interpretation should be “40 miles per U.K. gallon.” In U.K., one would think of gallon as U.K. gallon; however, in the U.S., the word “gallon” would naturally be interpreted as U.S. gallon. Note also that since 1 U.K. gallon 4.5460900 L and 1 U.S. gallon 3.7854118 L , the relationship between the two is 1 U.S. gallon 1 U.K. gallon (4.5460900 L) 1.20095 U.S. gallons 3.7854118 L ANALYZE (a) The amount of gasoline actually required is 750 miles 18.75 U.K. gallons 18.8 U.K. gallons 40 miles/U.K. gallon V This means that the driver mistakenly believes that the car should need 18.8 U.S. gallons. (b) Using the conversion factor found above, this is equivalent to 1.20095 U.S. gallons 18.75 U.K. gallons 22.5 U.S. gallons 1 U.K. gallon V LEARN One U.K. gallon is greater than one U.S gallon by roughly a factor of 1.2 in volume. Therefore, 40 mi/U.K. gallon is less fuel-efficient than 40 mi/U.S. gallon. 40. Equation 1-9 gives (to very high precision!) the conversion from atomic mass units to kilograms. Since this problem deals with the ratio of total mass (1.0 kg) divided by the mass of one atom (1.0 u, but converted to kilograms), then the computation reduces to simply taking the reciprocal of the number given in Eq. 1-9 and rounding off appropriately. Thus, the answer is 6.0 1026. 41. THINK This problem involves converting cord, a non-SI unit for volume, to SI unit. EXPRESS Using the (exact) conversion 1 in. = 2.54 cm = 0.0254 m for length, we have 17 0.0254 m 1 ft12 in (12 in.) 0.3048 m 1in . Thus, 3 3 31 ft (0.3048 m) 0.0283 m for volume (these results also can be found in Appendix D). ANALYZE The volume of a cord of wood is 3(8 ft) (4 ft) (4 ft) 128 ftV . Using the conversion factor found above, we obtain 3 3 3 3 3 0.0283 m 1cord 128 ft (128 ft ) 3.625 m 1 ft V which implies that 3 11 m cord 0.276 cord 0.3 cord 3.625 . LEARN The unwanted units ft 3 all cancel out, as they should. In conversions, units obey the same algebraic rules as variables and numbers. 42. (a) In atomic mass units, the mass of one molecule is (16 + 1 + 1)u = 18 u. Using Eq. 1-9, we find 27 261.6605402 10 kg18u = 18u 3.0 10 kg. 1u (b) We divide the total mass by the mass of each molecule and obtain the (approximate) number of water molecules: 21 46 26 1.4 10 5 10 . 3.0 10 N 43. A million milligrams comprise a kilogram, so 2.3 kg/week is 2.3 106 mg/week. Figuring 7 days a week, 24 hours per day, 3600 second per hour, we find 604800 seconds are equivalent to one week. Thus, (2.3 106 mg/week)/(604800 s/week) = 3.8 mg/s. 44. The volume of the water that fell is 2 2 2 6 2 6 3 1000 m 0.0254 m 26 km 2.0 in. 26 km 2.0 in. 1 km 1 in. 26 10 m 0.0508 m 1.3 10 m . V We write the mass-per-unit-volume (density) of the water as: 3 31 10 kg m . m V The mass of the water that fell is therefore given by m = V: 3 3 6 3 91 10 kg m 1.3 10 m 1.3 10 kg.m CHAPTER 1 18 45. The number of seconds in a year is 3.156 107. This is listed in Appendix D and results from the product (365.25 day/y) (24 h/day) (60 min/h) (60 s/min). (a) The number of shakes in a second is 10 8 ; therefore, there are indeed more shakes per second than there are seconds per year. (b) Denoting the age of the universe as 1 u-day (or 86400 u-sec), then the time during which humans have existed is given by 10 10 10 6 10 4 u - day, which may also be expressed as 10 86400 1 8 64 F HG I KJ u - day u - sec u -day u - sec.c h . 46. The volume removed in one year is V = (75 10 m ) (26 m) 2 10 m4 2 7 3 , which we convert to cubic kilometers: V F HG I KJ 2 10 1 0 0207 3 m km 1000 m km3 3c h . . 47. THINK This problem involves expressing the speed of light in astronomical units per minute. EXPRESS We first convert meters to astronomical units (AU), and seconds to minutes, using 81000 m 1 km, 1 AU 1.50 10 km, 60 s 1 min. ANALYZE Using the conversion factors above, the speed of light can be rewritten as 8 8 8 3.0 10 m 1 km AU 60 s 3.0 10 m/s 0.12 AU min. s 1000 m 1.50 10 km min c LEARN When expressed the speed of light c in AU/min, we readily see that it takes about 8.3 (= 1/0.12) minutes for sunlight to reach the Earth (i.e., to travel a distance of 1 AU). 48. Since one atomic mass unit is 241u 1.66 10 g (see Appendix D), the mass of one mole of atoms is about 24 23(1.66 10 g)(6.02 10 ) 1g.m On the other hand, the mass of one mole of atoms in the common Eastern mole is 19 75 g 10 g 7.5 m Therefore, in atomic mass units, the average mass of one atom in the common Eastern mole is 23 23 10 g 1.66 10 g 10 u. 6.02 10A m N 49. (a) Squaring the relation 1 ken = 1.97 m, and setting up the ratio, we obtain 1 1 197 1 3 88 2ken m m m 2 2 2 2 . . . (b) Similarly, we find 1 1 197 1 7 65 3 3ken m m m 3 3 3 . . . (c) The volume of a cylinder is the circular area of its base multiplied by its height. Thus, 22 33.00 5.50 156 ken .r h (d) If we multiply this by the result of part (b), we determine the volume in cubic meters: (155.5)(7.65) = 1.19 103 m3. 50. According to Appendix D, a nautical mile is 1.852 km, so 24.5 nautical miles would be 45.374 km. Also, according to Appendix D, a mile is 1.609 km, so 24.5 miles is 39.4205 km. The difference is 5.95 km. 51. (a) For the minimum (43 cm) case, 9 cubits converts as follows: 0.43m 9cubits 9cubits 3.9m. 1cubit And for the maximum (53 cm) case we have 0.53m 9cubits 9cubits 4.8m. 1cubit (b) Similarly, with 0.43 m 430 mm and 0.53 m 530 mm, we find 3.9 103 mm and 4.8 103 mm, respectively. (c) We can convert length and diameter first and then compute the volume, or first compute the volume and then convert. We proceed using the latter approach (where d is diameter and is length). CHAPTER 1 20 3 2 3 3 3 cylinder, min 0.43m 28 cubit 28 cubit 2.2 m . 4 1 cubit V d Similarly, with 0.43 m replaced by 0.53 m, we obtain Vcylinder, max = 4.2 m 3 . 52. Abbreviating wapentake as “wp” and assuming a hide to be 110 acres, we set up the ratio 25 wp/11 barn along with appropriate conversion factors: 2 28 2 36 100 hide 110 acre 4047 m 1 wp 1acre1 hide 1 10 m 1 barn 25 wp 1 10 . 11 barn 53. THINK The objective of this problem is to convert the Earth-Sun distance (1 AU) to parsecs and light-years. EXPRESS To relate parsec (pc) to AU, we note that when is measured in radians, it is equal to the arc length s divided by the radius R. For a very large radius circle and small value of , the arc may be approximated as the straight line-segment of length 1 AU. Thus, 6 1 arcmin 1 2 radian 1 arcsec 1 arcsec 4.85 10 rad 60 arcsec 60 arcmin 360 . Therefore, one parsec is 5 6 1 AU 1 pc 2.06 10 AU 4.85 10 s . Next, we relate AU to light-year (ly). Since a year is about 3.16 107 s, 7 121ly 186,000mi s 3.16 10 s 5.9 10 mi . ANALYZE (a) Since 51 pc 2.06 10 AU , inverting the relation gives 65 1 pc 1 AU 1 AU 4.9 10 pc. 2.06 10 AU (b) Given that 61AU 92.9 10 mi and 121 ly 5.9 10 mi , the two expressions together lead to 6 6 5 12 1ly 1AU 92.9 10 mi (92.9 10 mi) 1.57 10 ly 5.9 10 mi . 21 LEARN Our results can be further combined to give 1pc 3.2 ly. From the above expression, we readily see that it takes 51.57 10 y , or about 8.3 min, for Sunlight to travel a distance of 1 AU to reach the Earth. 54. (a) Using Appendix D, we have 1 ft = 0.3048 m, 1 gal = 231 in. 3 , and 1 in. 3 = 1.639 10 2 L. From the latter two items, we find that 1 gal = 3.79 L. Thus, the quantity 460 ft 2 /gal becomes 2 2 2 2460 ft 1 m 1gal460 ft /gal 11.3 m L. gal 3.28 ft 3.79 L (b) Also, since 1 m 3 is equivalent to 1000 L, our result from part (a) becomes 2 2 4 1 3 11.3 m 1000L 11.3 m /L 1.13 10 m . L 1 m (c) The inverse of the original quantity is (460 ft 2 /gal) 1 = 2.17 103 gal/ft2. (d) The answer in (c) represents the volume of the paint (in gallons) needed to cover a square foot of area. From this, we could also figure the paint thickness [it turns out to be about a tenth of a millimeter, as one sees by taking the reciprocal of the answer in part (b)]. 55. (a) The receptacle is a volume of (40 cm)(40 cm)(30 cm) = 48000 cm 3 = 48 L = (48)(16)/11.356 = 67.63 standard bottles, which is a little more than 3 nebuchadnezzars (the largest bottle indicated). The remainder, 7.63 standard bottles, is just a little less than 1 methuselah. Thus, the answer to part (a) is 3 nebuchadnezzars and 1 methuselah. (b) Since 1 methuselah.= 8 standard bottles, then the extra amount is 8 7.63 = 0.37 standard bottle. (c) Using the conversion factor 16 standard bottles = 11.356 L, we have 11.356 L 0.37 standard bottle (0.37 standard bottle) 0.26 L. 16 standard bottles 56. The mass of the pig is 3.108 slugs, or (3.108)(14.59) = 45.346 kg. Referring now to the corn, a U.S. bushel is 35.238 liters. Thus, a value of 1 for the corn-hog ratio would be equivalent to 35.238/45.346 = 0.7766 in the indicated metric units. Therefore, a value of 5.7 for the ratio corresponds to 5.7(0.777) 4.4 in the indicated metric units. 57. Two jalapeño peppers have spiciness = 8000 SHU, and this amount multiplied by 400 (the number of people) is 3.2 106 SHU, which is roughly ten times the SHU value for a CHAPTER 1 22 single habanero pepper. More precisely, 10.7 habanero peppers will provide that total required SHU value. 58. In the simplest approach, we set up a ratio for the total increase in horizontal depth x (where x = 0.05 m is the increase in horizontal depth per step) steps 4.57 0.05 m 1.2 m. 0.19 x N x However, we can approach this more carefully by noting that if there are N = 4.57/.19 24 rises then under normal circumstances we would expect N 1 = 23 runs (horizontal pieces) in that staircase. This would yield (23)(0.05 m) = 1.15 m, which - to two significant figures - agrees with our first result. 59. The volume of the filled container is 24000 cm 3 = 24 liters, which (using the conversion given in the problem) is equivalent to 50.7 pints (U.S). The expected number is therefore in the range from 1317 to 1927 Atlantic oysters. Instead, the number received is in the range from 406 to 609 Pacific oysters. This represents a shortage in the range of roughly 700 to 1500 oysters (the answer to the problem). Note that the minimum value in our answer corresponds to the minimum Atlantic minus the maximum Pacific, and the maximum value corresponds to the maximum Atlantic minus the minimum Pacific. 60. (a) We reduce the stock amount to British teaspoons: 1 6 2 2 24 breakfastcup = 2 8 2 2 = 64 teaspoons 1 teacup = 8 2 2 = 32 teaspoons 6 tablespoons = teaspoons 1 dessertspoon = 2 teaspoons which totals to 122 British teaspoons, or 122 U.S. teaspoons since liquid measure is being used. Now with one U.S cup equal to 48 teaspoons, upon dividing 122/48 2.54, we find this amount corresponds to 2.5 U.S. cups plus a remainder of precisely 2 teaspoons. In other words, 122 U.S. teaspoons = 2.5 U.S. cups + 2 U.S. teaspoons. (b) For the nettle tops, one-half quart is still one-half quart. (c) For the rice, one British tablespoon is 4 British teaspoons which (since dry-goods measure is being used) corresponds to 2 U.S. teaspoons. (d) A British saltspoon is 1 2 British teaspoon which corresponds (since dry-goods measure is again being used) to 1 U.S. teaspoon. 23 Chapter 2 1. The speed (assumed constant) is v = (90 km/h)(1000 m/km) (3600 s/h) = 25 m/s. Thus, in 0.50 s, the car travels a distance d = vt = (25 m/s)(0.50 s) 13 m. 2. (a) Using the fact that time = distance/velocity while the velocity is constant, we find avg 73.2 m 73.2 m 3.05 m1.22 m/s 73.2 m 73.2 m 1.74 m/s.v (b) Using the fact that distance = vt while the velocity v is constant, we find vavg m / s)(60 s) m / s)(60 s) s m / s. ( . ( . . 122 305 120 214 (c) The graphs are shown below (with meters and seconds understood). The first consists of two (solid) line segments, the first having a slope of 1.22 and the second having a slope of 3.05. The slope of the dashed line represents the average velocity (in both graphs). The second graph also consists of two (solid) line segments, having the same slopes as before — the main difference (compared to the first graph) being that the stage involving higher-speed motion lasts much longer. 3. THINK This one-dimensional kinematics problem consists of two parts, and we are asked to solve for the average velocity and average speed of the car. EXPRESS Since the trip consists of two parts, let the displacements during first and second parts of the motion be x1 and x2, and the corresponding time intervals be t1 and t2, respectively. Now, because the problem is one-dimensional and both displacements are in the same direction, the total displacement is simply x = x1 + x2, and the total time for the trip is t = t1 + t2. Using the definition of average velocity given in Eq. 2-2, we have 1 2 avg 1 2 . x xx v t t t CHAPTER 2 24 To find the average speed, we note that during a time t if the velocity remains a positive constant, then the speed is equal to the magnitude of velocity, and the distance is equal to the magnitude of displacement, with | | .d x v t ANALYZE (a) During the first part of the motion, the displacement is x1 = 40 km and the time taken is t1 40 133 ( . km) (30 km / h) h. Similarly, during the second part of the trip the displacement is x2 = 40 km and the time interval is t2 40 0 67 ( . km) (60 km / h) h. The total displacement is x = x1 + x2 = 40 km + 40 km = 80 km, and the total time elapsed is t = t1 + t2 = 2.00 h. Consequently, the average velocity is avg (80 km) 40 km/h. (2.0 h) x v t (b) In this case, the average speed is the same as the magnitude of the average velocity: avg 40 km/h.s (c) The graph of the entire trip, shown below, consists of two contiguous line segments, the first having a slope of 30 km/h and connecting the origin to (t1, x1) = (1.33 h, 40 km) and the second having a slope of 60 km/h and connecting (t1, x1) to (t, x) = (2.00 h, 80 km). From the graphical point of view, the slope of the dashed line drawn from the origin to (t, x) represents the average velocity. LEARN The average velocity is a vector quantity that depends only on the net displacement (also a vector) between the starting and ending points. 4. Average speed, as opposed to average velocity, relates to the total distance, as opposed to the net displacement. The distance D up the hill is, of course, the same as the distance down the hill, and since the speed is constant (during each stage of the 25 motion) we have speed = D/t. Thus, theaverage speed is D D t t D D v D v up down up down up down 2 which, after canceling D and plugging in vup = 40 km/h and vdown = 60 km/h, yields 48 km/h for the average speed. 5. THINK In this one-dimensional kinematics problem, we’re given the position function x(t), and asked to calculate the position and velocity of the object at a later time. EXPRESS The position function is given as x(t) = (3 m/s)t – (4 m/s2)t2 + (1 m/s3)t3. The position of the object at some instant t0 is simply given by x(t0). For the time interval 1 2t t t , the displacement is 2 1( ) ( )x x t x t . Similarly, using Eq. 2-2, the average velocity for this time interval is 2 1 avg 2 1 ( ) ( ) . x t x tx v t t t ANALYZE (a) Plugging in t = 1 s into x(t) yields x(1 s) = (3 m/s)(1 s) – (4 m/s2)(1 s)2 + (1 m/s3)(1 s)3 = 0. (b) With t = 2 s we get x(2 s) = (3 m/s)(2 s) – (4 m/s2) (2 s)2 + (1 m/s3)(2 s)3 = –2 m. (c) With t = 3 s we have x (3 s) = (3 m/s)(3 s) – (4 m/s2) (3 s)2 + (1 m/s3)(3 s)3 = 0 m. (d) Similarly, plugging in t = 4 s gives x(4 s) = (3 m/s)(4 s) – (4 m/s2)(4 s)2 + (1 m/s3) (4 s)3 = 12 m. (e) The position at t = 0 is x = 0. Thus, the displacement between t = 0 and t = 4 s is (4 s) (0) 12 m 0 12 m.x x x (f) The position at t = 2 s is subtracted from the position at t = 4 s to give the displacement: (4 s) (2 s) 12 m ( 2 m) 14 mx x x . Thus, the average velocity is avg 14 m 7 m/s. 2 s x v t (g) The position of the object for the interval 0 t 4 is plotted below. The straight line drawn from the point at (t, x) = (2 s, –2 m) to (4 s, 12 m) would represent the average velocity, answer for part (f). CHAPTER 2 26 LEARN Our graphical representation illustrates once again that the average velocity for a time interval depends only on the net displacement between the starting and ending points. 6. Huber’s speed is v0 = (200 m)/(6.509 s) =30.72 m/s = 110.6 km/h, where we have used the conversion factor 1 m/s = 3.6 km/h. Since Whittingham beat Huber by 19.0 km/h, his speed is v1 = (110.6 km/h + 19.0 km/h) = 129.6 km/h, or 36 m/s (1 km/h = 0.2778 m/s). Thus, using Eq. 2-2, the time through a distance of 200 m for Whittingham is 1 200 m 5.554 s. 36 m/s x t v 7. Recognizing that the gap between the trains is closing at a constant rate of 60 km/h, the total time that elapses before they crash is t = (60 km)/(60 km/h) = 1.0 h. During this time, the bird travels a distance of x = vt = (60 km/h)(1.0 h) = 60 km. 8. The amount of time it takes for each person to move a distance L with speed sv is / st L v . With each additional person, the depth increases by one body depth d (a) The rate of increase of the layer of people is (0.25 m)(3.50 m/s) 0.50 m/s / 1.75 m s s dvd d R t L v L (b) The amount of time required to reach a depth of 5.0 mD is 5.0 m 10 s 0.50 m/s D t R 9. Converting to seconds, the running times are t1 = 147.95 s and t2 = 148.15 s, respectively. If the runners were equally fast, then 1 2 avg avg1 2 1 2 . L L s s t t From this we obtain 27 2 2 1 1 1 1 1 148.15 1 1 0.00135 1.4 m 147.95 t L L L L L t where we set L1 1000 m in the last step. Thus, if L1 and L2 are no different than about 1.4 m, then runner 1 is indeed faster than runner 2. However, if L1 is shorter than L2 by more than 1.4 m, then runner 2 would actually be faster. 10. Let wv be the speed of the wind and cv be the speed of the car. (a) Suppose during time interval 1t , the car moves in the same direction as the wind. Then the effective speed of the car is given by ,1eff c wv v v , and the distance traveled is ,1 1 1( )eff c wd v t v v t . On the other hand, for the return trip during time interval t2, the car moves in the opposite direction of the wind and the effective speed would be ,2eff c wv v v . The distance traveled is ,2 2 2( )eff c wd v t v v t . The two expressions can be rewritten as 1 2 andc w c w d d v v v v t t Adding the two equations and dividing by two, we obtain 1 2 1 2 c d d v t t . Thus, method 1 gives the car’s speed cv a in windless situation. (b) If method 2 is used, the result would be 2 2 2 1 2 1 2 2 2 1 ( ) / 2 c w w c c c c c w c w v v vd d d v v d dt t t t v v v v v v . The fractional difference is 2 2 4(0.0240) 5.76 10c c w c c v v v v v . 11. The values used in the problem statement make it easy to see that the first part of the trip (at 100 km/h) takes 1 hour, and the second part (at 40 km/h) also takes 1 hour. Expressed in decimal form, the time left is 1.25 hour, and the distance that remains is 160 km. Thus, a speed v = (160 km)/(1.25 h) = 128 km/h is needed. 12. (a) Let the fast and the slow cars be separated by a distance d at t = 0. If during the time interval / (12.0 m) /(5.0 m/s) 2.40 sst L v in which the slow car has moved a distance of 12.0 mL , the fast car moves a distance of vt d L to join the line of slow cars, then the shock wave would remain stationary. The condition implies a separation of (25 m/s)(2.4 s) 12.0 m 48.0 m.d vt L (b) Let the initial separation at 0t be 96.0 m.d At a later time t, the slow and CHAPTER 2 28 the fast cars have traveled sx v t and the fast car joins the line by moving a distance d x . From , s x d x t v v we get 5.00 m/s (96.0 m) 24.0 m, 25.0 m/s 5.00 m/s s s v x d v v which in turn gives (24.0 m) /(5.00 m/s) 4.80 s.t Since the rear of the slow-car pack has moved a distance of 24.0 m 12.0 m 12.0 mx x L downstream, the speed of the rear of the slow-car pack, or equivalently, the speed of the shock wave, is shock 12.0 m 2.50 m/s. 4.80 s x v t (c) Since x L , the direction of the shock wave is downstream. 13. (a) Denoting the travel time and distance from San Antonio to Houston as T and D, respectively, the average speed is avg1 (55 km/h)( /2) (90 km/h)( / 2) 72.5 km/h D T T s T T which should be rounded to 73 km/h. (b) Using the fact that time = distance/speed while the speed is constant, we find avg2 / 2 / 2 55 km/h 90 km/h 68.3 km/h D D D D s T which should be rounded to 68 km/h. (c) The total distance traveled (2D) must not be confused with the net displacement (zero). We obtain for the two-way trip avg 72.5 km/h 68.3 km/h 2 70 km/h. D D D s (d) Since the net displacement vanishes, the average velocity for the trip in its entirety is zero. (e) In asking for a sketch, the problem is allowing the student to arbitrarily set the distance D (the intent is not to make the student go to an atlas to look it up); the student can just as easily arbitrarily set T instead of D, as will be clear in the following discussion. We brieflydescribe the graph (with kilometers-per-hour understood for the slopes): two contiguous line segments, the first having a slope of 55 and connecting the origin to (t1, x1) = (T/2, 55T/2) and the second having a slope of 90 and connecting (t1, x1) to (T, D) where D = (55 + 90)T/2. The average velocity, from the 29 graphical point of view, is the slope of a line drawn from the origin to (T, D). The graph (not drawn to scale) is depicted below: 14. Using the general property d dx bx b bxexp( ) exp( ) , we write v dx dt d t dt e t de dt t t F HG I KJ F HG I KJ ( ) ( ) 19 19 . If a concern develops about the appearance of an argument of the exponential (–t) apparently having units, then an explicit factor of 1/T where T = 1 second can be inserted and carried through the computation (which does not change our answer). The result of this differentiation is v t e t 16 1( ) with t and v in SI units (s and m/s, respectively). We see that this function is zero when t = 1 s. Now that we know when it stops, we find out where it stops by plugging our result t = 1 into the given function x = 16te –t with x in meters. Therefore, we find x = 5.9 m. 15. We use Eq. 2-4 to solve the problem. (a) The velocity of the particle is v dx dt d dt t t t ( ) .4 12 3 12 62 Thus, at t = 1 s, the velocity is v = (–12 + (6)(1)) = –6 m/s. (b) Since v 0, it is moving in the –x direction at t = 1 s. (c) At t = 1 s, the speed is |v| = 6 m/s. (d) For 0 t 2 s, |v| decreases until it vanishes. For 2 t 3 s, |v| increases from zero to the value it had in part (c). Then, |v| is larger than that value for t 3 s. (e) Yes, since v smoothly changes from negative values (consider the t = 1 result) to positive (note that as t + , we have v + ). One can check that v = 0 when 2 s.t CHAPTER 2 30 (f) No. In fact, from v = –12 + 6t, we know that v 0 for t 2 s. 16. We use the functional notation x(t), v(t), and a(t) in this solution, where the latter two quantities are obtained by differentiation: v t dx t dt t a t dv t dt b g b g b g b g 12 12and with SI units understood. (a) From v(t) = 0 we find it is (momentarily) at rest at t = 0. (b) We obtain x(0) = 4.0 m. (c) and (d) Requiring x(t) = 0 in the expression x(t) = 4.0 – 6.0t2 leads to t = 0.82 s for the times when the particle can be found passing through the origin. (e) We show both the asked-for graph (on the left) as well as the “shifted” graph that is relevant to part (f). In both cases, the time axis is given by –3 t 3 (SI units understood). (f) We arrived at the graph on the right (shown above) by adding 20t to the x(t) expression. (g) Examining where the slopes of the graphs become zero, it is clear that the shift causes the v = 0 point to correspond to a larger value of x (the top of the second curve shown in part (e) is higher than that of the first). 17. We use Eq. 2-2 for average velocity and Eq. 2-4 for instantaneous velocity, and work with distances in centimeters and times in seconds. (a) We plug into the given equation for x for t = 2.00 s and t = 3.00 s and obtain x2 = 21.75 cm and x3 = 50.25 cm, respectively. The average velocity during the time interval 2.00 t 3.00 s is v x t avg cm cm s s 50 25 2175 300 2 00 . . . . which yields vavg = 28.5 cm/s. (b) The instantaneous velocity is v tdx dt 4 5 2. , which, at time t = 2.00 s, yields v = (4.5)(2.00) 2 = 18.0 cm/s. 31 (c) At t = 3.00 s, the instantaneous velocity is v = (4.5)(3.00) 2 = 40.5 cm/s. (d) At t = 2.50 s, the instantaneous velocity is v = (4.5)(2.50) 2 = 28.1 cm/s. (e) Let tm stand for the moment when the particle is midway between x2 and x3 (that is, when the particle is at xm = (x2 + x3)/2 = 36 cm). Therefore, x t tm m m 9 75 15 2596 3. . . in seconds. Thus, the instantaneous speed at this time is v = 4.5(2.596) 2 = 30.3 cm/s. (f) The answer to part (a) is given by the slope of the straight line between t = 2 and t = 3 in this x-vs-t plot. The answers to parts (b), (c), (d), and (e) correspond to the slopes of tangent lines (not shown but easily imagined) to the curve at the appropriate points. 18. (a) Taking derivatives of x(t) = 12t 2 – 2t3 we obtain the velocity and the acceleration functions: v(t) = 24t – 6t2 and a(t) = 24 – 12t with length in meters and time in seconds. Plugging in the value t = 3 yields (3) 54 mx . (b) Similarly, plugging in the value t = 3 yields v(3) = 18 m/s. (c) For t = 3, a(3) = –12 m/s2. (d) At the maximum x, we must have v = 0; eliminating the t = 0 root, the velocity equation reveals t = 24/6 = 4 s for the time of maximum x. Plugging t = 4 into the equation for x leads to x = 64 m for the largest x value reached by the particle. (e) From (d), we see that the x reaches its maximum at t = 4.0 s. (f) A maximum v requires a = 0, which occurs when t = 24/12 = 2.0 s. This, inserted into the velocity equation, gives vmax = 24 m/s. (g) From (f), we see that the maximum of v occurs at t = 24/12 = 2.0 s. (h) In part (e), the particle was (momentarily) motionless at t = 4 s. The acceleration at that time is readily found to be 24 – 12(4) = –24 m/s2. CHAPTER 2 32 (i) The average velocity is defined by Eq. 2-2, so we see that the values of x at t = 0 and t = 3 s are needed; these are, respectively, x = 0 and x = 54 m (found in part (a)). Thus, vavg = 54 0 3 0 = 18 m/s. 19. THINK In this one-dimensional kinematics problem, we’re given the speed of a particle at two instants and asked to calculate its average acceleration. EXPRESS We represent the initial direction of motion as the +x direction. The average acceleration over a time interval 1 2t t t is given by Eq. 2-7: 2 1 avg 2 1 ( ) ( ) . v t v tv a t t t ANALYZE Let v1 = +18 m/s at 1 0t and v2 = –30 m/s at t2 = 2.4 s. Using Eq. 2-7 we find 22 1 avg 2 1 ( ) ( ) ( 30 m/s) ( 1m/s) 20 m/s 2.4 s 0 v t v t a t t . LEARN The average acceleration has magnitude 20 m/s 2 and is in the opposite direction to the particle’s initial velocity. This makes sense because the velocity of the particle is decreasing over the time interval. With 1 0t , the velocity of the particle as a function of time can be written as 2 0 (18 m/s) (20 m/s )v v at t . 20. We use the functional notation x(t), v(t) and a(t) and find the latter two quantities by differentiating: v t dx t t t a t dv t dt tb g b g b g b g 15 20 302 and with SI units understood. These expressions are used in the parts that follow. (a) From 0 15 202 t , we see that the only positive value of t for which the particle is (momentarily) stopped is t 20 15 12/ . s . (b) From 0 = – 30t, we find a(0) = 0 (that is, it vanishes at t = 0). (c) It is clear that a(t) = – 30t is negative for t > 0. (d) The acceleration a(t) = – 30t is positive for t < 0. (e) The graphs are shown below. SI units are understood. 33 21. We use Eq. 2-2 (average velocity) andEq. 2-7 (average acceleration). Regarding our coordinate choices, the initial position of the man is taken as the origin and his direction of motion during 5 min t 10 min is taken to be the positive x direction. We also use the fact that x v t ' when the velocity is constant during a time interval t ' . (a) The entire interval considered is t = 8 – 2 = 6 min, which is equivalent to 360 s, whereas the sub-interval in which he is moving is only 8 5 3min 180 s.t' His position at t = 2 min is x = 0 and his position at t = 8 min is x v t (2.2)(180) 396 m . Therefore, vavg m s m / s 396 0 360 110. . (b) The man is at rest at t = 2 min and has velocity v = +2.2 m/s at t = 8 min. Thus, keeping the answer to 3 significant figures, aavg 2 m / s s m / s 2 2 0 360 0 00611 . . . (c) Now, the entire interval considered is t = 9 – 3 = 6 min (360 s again), whereas the sub-interval in which he is moving is 9 5 4min 240 st ). His position at 3 mint is x = 0 and his position at t = 9 min is (2.2)(240) 528 mx v t . Therefore, vavg m s m / s. 528 0 360 147. (d) The man is at rest at t = 3 min and has velocity v = +2.2 m/s at t = 9 min. Consequently, aavg = 2.2/360 = 0.00611 m/s 2 just as in part (b). (e) The horizontal line near the bottom of this x-vs-t graph represents the man standing at x = 0 for 0 t < 300 s and the linearly rising line for 300 t 600 s represents his constant-velocity motion. The lines represent the answers to part (a) and (c) in the sense that their slopes yield those results. The graph of v-vs-t is not shown here, but would consist of two horizontal “steps” (one at v = 0 for 0 t < 300 s and the next at v = 2.2 m/s for 300 CHAPTER 2 34 t 600 s). The indications of the average accelerations found in parts (b) and (d) would be dotted lines connecting the “steps” at the appropriate t values (the slopes of the dotted lines representing the values of aavg). 22. In this solution, we make use of the notation x(t) for the value of x at a particular t. The notations v(t) and a(t) have similar meanings. (a) Since the unit of ct 2 is that of length, the unit of c must be that of length/time 2 , or m/s 2 in the SI system. (b) Since bt 3 has a unit of length, b must have a unit of length/time 3 , or m/s 3 . (c) When the particle reaches its maximum (or its minimum) coordinate its velocity is zero. Since the velocity is given by v = dx/dt = 2ct – 3bt2, v = 0 occurs for t = 0 and for t c b 2 3 2 30 3 2 0 10 ( . ) ( . ) . m / s m / s s . 2 3 For t = 0, x = x0 = 0 and for t = 1.0 s, x = 1.0 m > x0. Since we seek the maximum, we reject the first root (t = 0) and accept the second (t = 1s). (d) In the first 4 s the particle moves from the origin to x = 1.0 m, turns around, and goes back to x( ( . )( . ( . )( .4 30 4 0 2 0 4 0 802 s) m / s s) m / s s) m .2 3 3 The total path length it travels is 1.0 m + 1.0 m + 80 m = 82 m. (e) Its displacement is x = x2 – x1, where x1 = 0 and x2 = –80 m. Thus, 80 mx . The velocity is given by v = 2ct – 3bt2 = (6.0 m/s2)t – (6.0 m/s3)t2. (f) Plugging in t = 1 s, we obtain 2 3 2(1 s) (6.0 m/s )(1.0 s) (6.0 m/s )(1.0 s) 0.v (g) Similarly, 2 3 2(2 s) (6.0 m/s )(2.0 s) (6.0 m/s )(2.0 s) 12m/s .v (h) 2 3 2(3 s) (6.0 m/s )(3.0 s) (6.0 m/s )(3.0 s) 36 m/s .v (i) 2 3 2(4 s) (6.0 m/s )(4.0 s) (6.0 m/s )(4.0 s) 72 m/s .v The acceleration is given by a = dv/dt = 2c – 6b = 6.0 m/s2 – (12.0 m/s3)t. (j) Plugging in t = 1 s, we obtain 2 3 2(1 s) 6.0 m/s (12.0 m/s )(1.0 s) 6.0 m/s .a (k) 2 3 2(2 s) 6.0 m/s (12.0 m/s )(2.0 s) 18 m/s .a 35 (l) 2 3 2(3 s) 6.0 m/s (12.0 m/s )(3.0 s) 30 m/s .a (m) 2 3 2(4 s) 6.0 m/s (12.0 m/s )(4.0 s) 42 m/s .a 23. THINK The electron undergoes a constant acceleration. Given the final speed of the electron and the distance it has traveled, we can calculate its acceleration. EXPRESS Since the problem involves constant acceleration, the motion of the electron can be readily analyzed using the equations given in Table 2-1: 0 2 0 0 2 2 0 0 (2 11) 1 (2 15) 2 2 ( ) (2 16) v v at x x v t at v v a x x The acceleration can be found by solving Eq. 2-16. ANALYZE With 5 0 1.50 10 m/sv , 65.70 10 m/sv , x0 = 0 and x = 0.010 m, we find the average acceleration to be 2 2 6 2 5 2 15 20 (5.7 10 m/s) (1.5 10 m/s) 1.62 10 m/s . 2 2(0.010 m) v v a x LEARN It is always a good idea to apply other equations in Table 2-1 not used for solving the problem as a consistency check. For example, since we now know the value of the acceleration, using Eq. 2-11, the time it takes for the electron to reach its final speed would be 6 5 90 15 2 5.70 10 m/s 1.5 10 m/s 3.426 10 s 1.62 10 m/s v v t a Substituting the value of t into Eq. 2-15, the distance the electron travels is 2 5 9 15 2 9 2 0 0 1 1 0 (1.5 10 m/s)(3.426 10 s) (1.62 10 m/s )(3.426 10 s) 2 2 0.01 m x x v t at This is what was given in the problem statement. So we know the problem has been solved correctly. 24. In this problem we are given the initial and final speeds, and the displacement, and are asked to find the acceleration. We use the constant-acceleration equation given in Eq. 2-16, v 2 = v 2 0 + 2a(x – x0). (a) Given that 0 0v , 1.6 m/s,v and 5.0 m,x the acceleration of the spores during the launch is CHAPTER 2 36 2 2 2 5 2 40 6 (1.6 m/s) 2.56 10 m/s 2.6 10 2 2(5.0 10 m) v v a g x (b) During the speed-reduction stage, the acceleration is 2 2 2 3 2 20 3 0 (1.6 m/s) 1.28 10 m/s 1.3 10 2 2(1.0 10 m) v v a g x The negative sign means that the spores are decelerating. 25. We separate the motion into two parts, and take the direction of motion to be positive. In part 1, the vehicle accelerates from rest to its highest speed; we are given v0 = 0; v = 20 m/s and a = 2.0 m/s 2 . In part 2, the vehicle decelerates from its highest speed to a halt; we are given v0 = 20 m/s; v = 0 and a = –1.0 m/s 2 (negative because the acceleration vector points opposite to the direction of motion). (a) From Table 2-1, we find t1 (the duration of part 1) from v = v0 + at. In this way, 120 0 2.0t yields t1 = 10 s. We obtain the duration t2 of part 2 from the same equation. Thus, 0 = 20 + (–1.0)t2 leads to t2 = 20 s, and the total is t = t1 + t2 = 30 s. (b) For part 1, taking x0 = 0, we use the equation v 2 = v 2 0 + 2a(x – x0) from Table 2-1 and find 2 2 2 2 0 2 (20 m/s) (0) 100 m 2 2(2.0 m/s ) v v x a . This position is then the initial position for part 2, so that when the same equation is used in part 2 we obtain 2 2 2 2 0 2 (0) (20 m/s) 100 m 2 2( 1.0 m/s ) v v x a . Thus, the final position is x = 300 m. That this is also the total distance traveled shouldbe evident (the vehicle did not "backtrack" or reverse its direction of motion). 26. The constant-acceleration condition permits the use of Table 2-1. (a) Setting v = 0 and x0 = 0 in 2 2 0 02 ( )v v a x x , we find 2 6 2 0 14 1 1 (5.00 10 ) 0.100 m . 2 2 1.25 10 v x a Since the muon is slowing, the initial velocity and the acceleration must have opposite signs. (b) Below are the time plots of the position x and velocity v of the muon from the moment it enters the field to the time it stops. The computation in part (a) made no reference to t, so that other equations from Table 2-1 (such as v v at 0 and 37 x v t at 0 1 2 2 ) are used in making these plots. 27. We use v = v0 + at, with t = 0 as the instant when the velocity equals +9.6 m/s. (a) Since we wish to calculate the velocity for a time before t = 0, we set t = –2.5 s. Thus, Eq. 2-11 gives v ( . . ( . .9 6 32 2 5 16 m/ s) m/ s s) m/ s.2c h (b) Now, t = +2.5 s and we find v ( . . ( .9 6 32 2 5 18 m/ s) m/ s s) m/ s.2c h 28. We take +x in the direction of motion, so v0 = +24.6 m/s and a = – 4.92 m/s 2 . We also take x0 = 0. (a) The time to come to a halt is found using Eq. 2-11: 0 2 24.6 m/s 0 5.00 s 4.92 m/s .v at t (b) Although several of the equations in Table 2-1 will yield the result, we choose Eq. 2-16 (since it does not depend on our answer to part (a)). 2 2 0 2 (24.6 m/s) 0 2 61.5 m 2 4.92 m/s .v ax x (c) Using these results, we plot 21 0 2 v t at (the x graph, shown next, on the left) and v0 + at (the v graph, on the right) over 0 t 5 s, with SI units understood. CHAPTER 2 38 29. We assume the periods of acceleration (duration t1) and deceleration (duration t2) are periods of constant a so that Table 2-1 can be used. Taking the direction of motion to be +x then a1 = +1.22 m/s 2 and a2 = –1.22 m/s 2 . We use SI units so the velocity at t = t1 is v = 305/60 = 5.08 m/s. (a) We denote x as the distance moved during t1, and use Eq. 2-16: 2 2 2 0 1 2 (5.08 m/s) 2 2(1.22 m/s ) v v a x x 10.59 m 10.6 m. (b) Using Eq. 2-11, we have 0 1 2 1 5.08 m/s 4.17 s. 1.22 m/s v v t a The deceleration time t2 turns out to be the same so that t1 + t2 = 8.33 s. The distances traveled during t1 and t2 are the same so that they total to 2(10.59 m) = 21.18 m. This implies that for a distance of 190 m – 21.18 m = 168.82 m, the elevator is traveling at constant velocity. This time of constant velocity motion is t3 16882 508 3321 . . . m m / s s. Therefore, the total time is 8.33 s + 33.21 s 41.5 s. 30. We choose the positive direction to be that of the initial velocity of the car (implying that a < 0 since it is slowing down). We assume the acceleration is constant and use Table 2-1. (a) Substituting v0 = 137 km/h = 38.1 m/s, v = 90 km/h = 25 m/s, and a = –5.2 m/s 2 into v = v0 + at, we obtain t 25 38 52 2 5 2 m / s m / s m / s s . . . (b) We take the car to be at x = 0 when the brakes are applied (at time t = 0). Thus, the coordinate of the car as a function of time is given by 2 2 1 38 m/s 5.2 m/s 2 x t t in SI units. This function is plotted from t = 0 to t = 2.5 s on the graph to the right. We have not shown the v-vs-t graph here; it is a descending straight line from v0 to v. 31. THINK The rocket ship undergoes a constant acceleration from rest, and we want to know the time elapsed and the distance traveled when the rocket reaches a certain speed. 39 EXPRESS Since the problem involves constant acceleration, the motion of the rocket can be readily analyzed using the equations in Table 2-1: 0 2 0 0 2 2 0 0 (2 11) 1 (2 15) 2 2 ( ) (2 16) v v at x x v t at v v a x x ANALYZE (a) Given that 29.8 m/sa , 0 0v and 70.1 3.0 10 m/sv c , we can solve 0v v at for the time: 7 60 2 3.0 10 m/s 0 3.1 10 s 9.8 m/s v v t a which is about 1.2 months. So it takes 1.2 months for the rocket to reach a speed of 0.1c starting from rest with a constant acceleration of 9.8 m/s 2 . (b) To calculate the distance traveled during this time interval, we evaluate x x v t at 0 0 1 2 2 , with x0 = 0 and 0 0v . The result is 2 6 2 13 1 9.8 m/s (3.1 10 s) 4.6 10 m. 2 x LEARN In solving parts (a) and (b), we did not use Eq. (2-16): 2 2 0 02 ( )v v a x x . This equation can be used to check our answers. The final velocity based on this equation is 2 2 13 7 0 02 ( ) 0 2(9.8 m/s )(4.6 10 m 0) 3.0 10 m/sv v a x x , which is what was given in the problem statement. So we know the problems have been solved correctly. 32. The acceleration is found from Eq. 2-11 (or, suitably interpreted, Eq. 2-7). a v t F HG I KJ 1020 1000 3600 14 202 4 2 km / h m / km s / h s m / s b g . . . In terms of the gravitational acceleration g, this is expressed as a multiple of 9.8 m/s 2 as follows: 2 2 202.4 m/s 21 . 9.8 m/s a g g 33. THINK The car undergoes a constant negative acceleration to avoid impacting a barrier. Given its initial speed, we want to know the distance it has traveled and the time elapsed prior to the impact. CHAPTER 2 40 EXPRESS Since the problem involves constant acceleration, the motion of the car can be readily analyzed using the equations in Table 2-1: 0 2 0 0 2 2 0 0 (2 11) 1 (2 15) 2 2 ( ) (2 16) v v at x x v t at v v a x x We take x0 = 0 and v0 = 56.0 km/h = 15.55 m/s to be the initial position and speed of the car. Solving Eq. 2-15 with t = 2.00 s gives the acceleration a. Once a is known, the speed of the car upon impact can be found by using Eq. 2-11. ANALYZE (a) Using Eq. 2-15, we find the acceleration to be 20 2 2 2 (24.0 m) (15.55 m/s)(2.00 s)2( ) 3.56m/s , (2.00 s) x v t a t or 2| | 3.56 m/sa . The negative sign indicates that the acceleration is opposite to the direction of motion of the car; the car is slowing down. (b) The speed of the car at the instant of impact is 2 0 15.55 m/s ( 3.56 m/s )(2.00 s) 8.43 m/sv v at which can also be converted to 30.3 km/h. LEARN In solving parts (a) and (b), we did not use Eq. 1-16. This equation can be used as a consistency check. The final velocity based on this equation is 2 2 2 0 02 ( ) (15.55 m/s) 2( 3.56 m/s )(24 m 0) 8.43 m/sv v a x x , which is what was calculated in (b). This indicates that the problems have been solved correctly. 34. Let d be the 220 m distance between the cars at t = 0, and v1 be the 20 km/h = 50/9 m/s speed (corresponding to a passing point of x1 = 44.5 m) and v2 be the 40 km/h =100/9 m/s speed (corresponding to a passing point of x2 = 76.6 m) of the red car. We have two equations (based on Eq. 2-17): d – x1 = vo t1 + 1 2 a t1 2
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