_________PROBLEMS

Section 3.1 The Vector Description of Motion

1. You walk west 220 m, then turn 450 toward the north and walk another 50 m. How far and in what direc­tion from your starling point do you end up? Solve both with a diagram and by adding vector compo­

-Qents. 2. For the vectors shown in Fig. 3-33, evaluate the vec­

tors A+B. A- B, A+C, A+B +C.

Fig. 3-33 Vectors for Problem 2; lengths ~re in arbitrary units.

3. A vector A is 10 units long and points 30° counter­clockwise (CCW) from horizontal. What are its x and y components on a coordinate system (0) with the x­axis horizontal and the v-axis vertical; (b) with the x­axis at 450 CCW from ilorizonta! and the y-axis 45° CCW from vertical; and (c) with the x-axis at 300 CCW from horizontal and the y-axis 900 CCW from the x­axis?

4. Express the sum of the unit vectors i, i, and k in unit vector notation, and deterrlline its magnitude.

5. The vector A is 12 units long and points 300 north of east. The vector B is 18 units long and points 45° west

7::\ of north. Find a vector C such that + B + C= o. \... 6J In a Hawaiian fool race, runners travel a twisting 58­

km path, climbing from the ocean shore to an altitude of 3300 m at the summit of Haleakala volcano. They wind up a horizontal distance 15 k.m northwest of their starting point. (0) At what angle must someone at the starting line beam a laser to send a signal to someone at the finishing line? (b) Moving at the speed of light, how long does it take the laser beam to go

) from start to finish line? 7. Let A=15i-40J and B=31j+18k. Find a vector C

-", such that A B+ C = O. 8. A mountain expedition starts a base camp at an alti­

tude of 5500 m. Four climbers then establish an ad­vance camp at an alHtude of 7400 m; the advance camp is southeast of the base camp, at a horizontal di tance of 8,2 km. From the advance camp, two climbers head directly north to a 8900-m summit, a horizontal distance of 2.1 km. Using a coordinate sys­tem with the x-axis eastward. the y-axis northward. and the z-axis upward. and with origin at the base

&

_

camp, express the positions of the advance camp and summit in unit vector notation. and determine the straight-line distance from bilse camp to summit.

9. An object i moving at 18 mis at an angle of 220° counterclockwise from the x-axis. What are the x- and y-component of its velocity?

10. A ar drives north at 40 milh for 10 min, then turns east and goes 5.0 lUi at 60 mi/h. Finally, it goes south­west at 30 mi/h for 6.0 min. Draw a vector diagram and determine (0) the car's displacement and (b) its average velocity for this trip.

11. For the car of the preceding problem, determine the average acceleration between the end of the first leg and tbe beginning of the third leg of the trip.

12. A biologist studying the motion of bacteria notes u bacterium at position r,=2.2i+3.7j-1.2k,um (l ,urn = 10- 6 Am). 6.2 s later the bacterium is at r2 = 4.6f + 1.9k ,um. What is its average velocity? Ex­press in unit vector notation, and calculate the mag­nitude.

1~A supersonic aircraft is travelling east at 2100 kin/h. It tben begins to turn southward. emerging from the turn 2.5 min later heading due south at 1800 kmIh. What are the magnitude and direction of its average acceleration during the turn?

14. For the preceding problem, write tbe initial and final velocities in uuH vector notation, and from them cal­culate the a' erage acceleration in unit vector uola­tion. Verify that your result is consistent with the an­swer to the preceding problem.

(1 An object undergoes acceleration of 2.31 + 3.6j m/s2

over a 10-s interval. At tbe end of this time, its velo . ity is 33i + 151 mls. (0) What was its velocity at the beginning of the 10-s interval? (b) By how much did its speed change? (c) By how much did its direction change? (d) Show that the speed change is not given by the magnitude of the acceleration times the time. Why not?

16. An object's position as a f!-lnction of time is given by r = (bt + ct)i + de)+ (el + j)k, where b, c, d, e, and f are constants. DetermiD the velodtv and ac elera­tion as functions of time. ­

17."' The position of an object is given by r=(cl-beH+dI2j, where c=6.7 mis, b=0.81 mls3

,

and d = 4. mis2• (a) Determine the object's velocil at

time 1=0. (b) How long does it take for the direction of motion to change by gOO? (c) By how much does the speed change during this time?

Section 3.2 ConstaIl! Acceleration

18. An airplane heads northeastward down a runway, ac­celerating from rest at the rate of 2.1 mls2

. Express the plane's velocity and position at t = 30 in unit vector notation. using a coordinate sy tern with x-axis east­ward and y-axis northwaJ"d, and with origin at lhe start of the plane's take-off roll.

66 3 Motion in More Than One Dimension

19. particle leaves the ongm with initial velocit r

0= 11 'i + 141 m/s. It undergoes a constant acceleration given by a= -1.2i+0.26) m/s2.) (0) When does the particle cross the y-axis? (b) What is its y-coordinate t that time? (c) How fast is it moving, and in what

direction, at that time? 20. Use Equation 3-11, along with the velocity calculated

in the first part of Example 3-7, to determine the final position of the rocket in that example.

21, Figure 3-34 shows a cathode-ra tube, used to dis­play electrical signals in oscilloscopes and other sci­entific instruments. Electrons are accelerated by the electron gun, then move down the center of the tube at 2.0 x 109 cm/s. In the 4.2-cm-long deflecting region they undergo an acceleration directed perpendicular to the Ion axis of the tube. The acceleration "steer" th m to a particular spot on ilie screen, where they produce a visible glow. (0) What acceleration is needed to deflect the electrons through 15°, as shown in the figure? (b) What is the shape of an electron's path in tile deflecting region?

Electron gun

Fig.3-34 A cathode-ray tube (Problem 21).

22. A particle starts from the ongm with initial velo­city Vo = vo'! and constant acceleration a = oj. Show that the particle's distance from the origin and its direction relative to the x-axis are given by d=IVv02+!02e and 6=tan'"'(aUZvo)'

Section 3.3 Projectile Motion

23. A carpenter tosses a shingle off a 9.4-m-high roof, giving it an initially horizontal velocity of 7.2 mls. (0) How long does it take to reach the ground? (b) Ho',v far does it move horizontally in this time?

24. You are trying to .roll a ball off a BO.O-cm-high table to quash a bug on the floor 0.0 cm from the table's edge. How fast should you roll the ball?

25. Repeat the precedin' problem, now with the bug moving away from the table at 30 mmls and 50.0 cm from the table when the ball leaves ilie table edge.

26. Mike is standing outside the physics building, 15 ft from ilie wall. Debbie, at a window 12 ft above, tosses a physics book horizontally. What speed should she give it if it is to reach Mike?

27. You are on the ground 3.0 m from the wall of a build­ing, and want to throw a package from your 1.5-m shoulder level to someone in a second floor window

4.2 m above the ground. At what speed and angle should you throw it so it just barely rear.:hes the \Nin­dow?

28. Derive a general formula for the horizontal distance covered by a projectile launched horizontally at speed \'0 from a height h.

29. A submarine-launched missile has a range of 4500 km. (0) Assuming the missile reaches its full speed within a short distance of the ocean surface, what speed is needed for this range when the missile is launched at a 45° angle? (b) What is the total flight time for the missile'? (The shortness of your answer is one reason that your survival in the nuclear age hangs in such precarious balance.) (c) To avoid destrur.:tion by space-based anUmissile weapons, the missile's tra­jectory can be "depressed" by giving it a lower launch angle. What is the minimum launch speed needed to give the same 4500-km range if til launch angle is ZOO? eglect air resistance and the curvature of the earth.

r3CJ:'-..A rescue airplane is Dying horizontally at speed Vo at an alUtude h above the ocean, attempting to drop a package of medical supplies to shipwreck victims in a lifeboat. At what line-of-sight angle 0. (Fig. '3-35) should the pilot release the package?

Fig. 3-35 Problem 30

31. At a circus. a human cannonbaJl is shot from a can­non at 35 km/h at an angle of 40°. U he leaves the cannon 1.0 ill off the ground. and lands in a net 2.0 m off the ground, how long is he in the air?

3Z: As you stand on the rim of ilie Grand Canyon, a friend 10 rn back from the rim throws you a baseball at 60 k:m/h and an initial angle of 10°. You miss ilie ball. and it winds up dropping 500 III below !Jle rim before it hits a trail descending into the canyon. Ne­glecting air resistance, (0) what horizontal distance from the rim would it travel and (b) at what speed would it hit ilie trail? (Neglect of air resistance here

is actually not justified; with air resi tance, your an­swers would be quite different.)

33. If you can hit a golf ball 180 In on earth, how far can you hit it on the moon? (Your answer is an underes­timate, because the distance on earth is restricted by air resistance as well as by a larger g.)

:n. Prove that a projectile launched on level ground reaches its maximum height midway along its trajec­tory.

35. A projectile launched at an angle 8 to the horizontal reaches a maximum height h. Show that its horizon­tal range is 411/tan8.

36. What is the minimum speed for throwing a ball over the house of Fig. 3-36 if the thrower is 15 ft from the wall? Assume the ball is thrown from essentially

to ground level.

45'

I10 II

1+------ 20 It---+1

f' 'r.J \/'{o Fig.3-36 Problem 36 "l:;. a 37. A circular fountain has jets of water directed fTom the

circumference inward at an angle of 45°. Each jet reaches a maximum height of 2.2 m. (0) [f all the jets converge in the center of the circle and at their initial heigbt, what is the radius of the fountain'! (b) If one of the jets is aimed at 100 too low, how far short of the center does it fall?

38. When the Olympics were held in Mexico City in 1968, many sports fans feared that the high altitude would result in poor athletic perform811 es due to re­duced oxygen. To their surpri e, new records were set in track and field events, probably as a result of lowered air resistance and a decrease in g to 9.786 m/s2

, both ultimately associated with the high alti­tude. In particular, Robert Beamon set a new world record of 8.90 m in the long jump. Photographs sug­gest that Beamon sti:lrled his jump at a 25° angle to the horizontal. ff he had jumped at sea level, where g= 9.81 m/s2

• at the same angle and initial speed as in Mexico City, hoVl far would Beamon have gone? c­glect air resistance in both cases (although its effect is actually more significant than UlC change in g).

39. Show that, for a given initial speed, the horizontal range of a projectile is the same for launch angles 45°+0'. and 45°-.0'., where 0: is between 0 and 45°.

40. A typical intercontinental ballistic lUi sile (ICBM) has a range of 10,000 kill and reaches a maximum altitude

Problems 67

of 1200 km. Assuming its rocket engine fires for only a short time at the beginning of the flight, determine (0) its launch angle and (b) its flight time. Neglect the curvature of tlle earth (tills is not a particularly good approximation here).

41. A basketball player is 15 ft horizontally from the cen­ter of the basket, which is 10 ft off the ground. At what angle should th player aim the ball if it is thrown from a height of 8.2 ft with a speed of 26 ft/s?

Section 3.4 Circular Motion

42. How fast would a car bave to round a turn 50 m in radius in order for its acceleration to be numericallv equal to that of gravity? .

43. Estimate the acceleration of the moon, which com­pletes a nearly circular orbit of 390,000 km radius in 27 cla) .

44. An object is in uniform circular motion. Make a graph of its average acceleration, measured in units of Vir, versus angular separation for points on the circular path spaced 40 . 30°, 20°, and 10° apart. Your graph should show the average acceleration approaching the instantaneous value v 2/r as the angular separation decreases.

45. When Apollo astronauts landed on the moon, they left one astronaut behind in a circular orbit around the moon. For the half of each orbit that took him around the back side of the moon, this astronaut was "the loneliest man in tJle universe," who could nei­ther see earth nor communicate by radio with earth or with his fellow astronauts on the moon. Estimate the duration of this total separation hom the re t of humanity, assuming a sufficiently low orbit that you can neglecl the distance above the lunar surface (see Appendix F for the acceleration of the moon's grav­ity).

46. lO-in-diam ter circular saw blade rotates at 3500 revolutions per minute. What is the acceleration of one of the sa teeth? Compare with the ace leration of gTavil~ .

47. A beetle can clin to a phonograph record only if the acceleration is less than 0.25 g. How far from the cen­ter of a 33!-rpm record can the beetle stand?

48. A jet is diving vertically downward at 1200 kmlh (see fig. 3-37). [f the pilot can withstand a maximum ae-

I v~t

L Fig.3-37 Problem 48

68 3 Motion in More Than One Dimension

celeration of 5g before losing consciousness. how close to the ground can the plane get before starting a quarter turn to pull out of the dive? Assume the plane's speed remains constant.

49. Electrons in a TV tube are deflected t.hrough a 550 an­gle as shown in Fig. 3-38. During the deflection they move at constant speed in a circular path of radius 4.30 em. [f they experience an acceleration of 3.35 x 10)7 mJs2

• how laD docs the deflection take?

r ~ 4.30 em

Fig. 3-38 Probl"rn 49

50. How fast would the earth have to be rotating for the radial acceleration of an object on the equator to equal g? Compare with earth's actual rotation rate.

51. t\ record turntable takes 14 s to reach full speed of 33~ revolutions per minute. \Nith constant tangential acceleration. what are the tangential and radial com­poncnts of thc acceleration of a point On the rim of a 12-in record 10 s after !lIe turntable is turned on?

52. 1\ plane is heading northward when it begins to turn eastward on a circular path of radius 9.10 km. At the instant it begins to turo, its acceleration vector points 22.0° north of east and has magnitude 2.60 mJs2

.

(0) What is the plane's speed? (b) At what rate is its speed increasing?

Section 3.5 Motion Is Relalive

53. 1\ dog paces around the perimeter of a rectangular barge that is headed up a river at 14 kmIh relative to the riverbank. The current in the water is at 3.0 kmIh. If the dog walks at 4.0 krnJh, what are its speeds rel­ative to (0) the shore and (b) the waler as it walks around the barge?

54. A jelliuer with an airspeed of 1000 km/h set' out on a 1500-km flight due south. To maintain a southward direction. however, the fJlane must be pointed 15° west of south. ff the flight takes 100 min. what is ilie wind velocity?

55. A car is moving at a constant 19 m/s. Its 72-cm-di­ameter wheels rotate at 8.4 revolutions per second.

What are the instantaneous velocity and acceleration of (a) the bottom of the wheel, (b) the top of the wheel. and (c) tbe forward edge of the wheel? Deter­mine these quantities both in ilie car's frame of ref­erence and in the ground frame of reference.

Supplementary Problems

56. A juggler's hands are 80 cm apart, and the balls being juggled reach a maximum height of 100 em above Ihe juggler's hands. (a) At what velocity do the balls leave the juggler's hands? (b) ff four balls are being juggled. how often must the juggler catch a ball?

57. t\ monkey is hanging from a branch a dislance 11 above the ground. A hunter stands a horizontal dis­tance d frOIll a point directly below tbe monkey. The hunter wants to shoot the monkey, but knows that the monkey will let go of the branch at ilie instant the gun fires. (0) Show that the bullet will hit the falling rnonkey if the gun is aimed directly at lhe monkey's initial position. (b) Your result in (a) is independent of the speed of the bullet as it leaves the gun. pro­vided Illat speed is above a certain minimum value that will ensure ilie bullet's reaching the monkey be­fore the monkey hits the ground. Show that this min­imum value is V(d 2 +h2)gl2h.

58. A child losses a ball over a flat-roofed house 3.2 m high and 7.4 m wide, so that it just clears the corners on bolh sides, as shown in Fig. 3-39. If the child stands 2.1 m from the wall. what are the ball's initial speed and launch angle? Assume the ball is launched essentially from ground lev I.

Fig. 3-39 Problem 58

59. A diver leaves a 3-m boaxd on a lrajectory that takes her 2.5 m above lhe board, and then into the wflter a horizontal distance of 2.8 III from the end of the board. At what sp ed and angle did she leave the board?

60. An object moves at constant speed v in the x-y plane, describing a circle of radius r centered at the origin. It is on the positive x-a;ds at time 1=0. Show that the position of the object as a function of time can be written r = r[cos(vt/r)i + sin(vtJr)il. where lhe argument of the sine and cosine is in radians. Differentiate this expression on e to obtain an expression for the veloc­ity and again for the acceleration. Show that ilie ac­celenltion has magnitude v 2fr and is directed radially inward (that is. opposite to r).