The Language of Physcs Knematcs The branch of mechancs that descrbes the moton of a body wthout regard to the cause of that moton (p. 39). Average velocty The average rate at whch the dsplacement vector changes wth tme. Snce a dsplacement s a vector, the velocty s also a vector (p. 39). Average speed The dstance that a body moves per unt tme. Speed s a scalar quantty (p. 4). Constant velocty A body movng n one drecton n such a way that t always travels equal dstances n equal tmes (p. 42). Acceleraton The rate at whch the velocty of a movng body changes wth tme (p. 43). Instantaneous velocty The velocty at a partcular nstant of tme. It s defned as the lmt of the rato of the change n the dsplacement of the body to the change n tme, as the tme nterval approaches zero. The magntude of the nstantaneous velocty s the nstantaneous speed of the movng body (p. 45). Knematc equatons of lnear moton A set of equatons that gves the dsplacement and velocty of the movng body at any nstant of tme, and the velocty of the movng body at any dsplacement, f the acceleraton of the body s a constant (p. 46). Freely fallng body Any body that s movng under the nfluence of gravty only. Hence, any body that s dropped or thrown on the surface of the earth s a freely fallng body (p. 5). Acceleraton due to gravty If ar frcton s gnored, all objects that are dropped near the surface of the earth, are accelerated toward the center of the earth wth an acceleraton of 9.80 m/s2 or 32 ft/s2 (p. 52). Projectle moton The moton of a body thrown or fred wth an ntal velocty v n a gravtatonal feld (p. 55). Trajectory The path through space followed by a projectle (p. 56). Range of a projectle The horzontal dstance from the pont where the projectle s launched to the pont where t returns to ts launch heght (p. 64). Summary of Important Equatons Average velocty = = *"2 ~ f VavE At h ~ h Acceleraton Av v v0 a = A7 = ^- (3.32) (3.33) Instantaneous velocty n two or more drectons, whch s a generalzaton of the nstantaneous velocty n one dmenson v = hm *- Al (3.8).. Ax v = hm AI^O Al Velocty at any tme v = v0 + ar Dsplacement at any tme (3.35) (3.34) Velocty at any dsplacement n the x-drecton u 2 = v ] + 2 a x ( 3. 6 ) Velocty at any dsplacement n the ^-drecton o2 = vl + 2ay For Projectle Moton x-dsplacement x = vvxt (3.6) (3.38) ^-dsplacement y = v0yt - \gt2 'x-component of velocty vx = v0x ^-component of velocty vy = %, - gt ^-component of velocty at any heght y 2 v2, = )2.,, 2gy y " O j / Range (3.39) (3.40) (3.4) (3.48) (3.47) Questons for Chapter 3. Dscuss the dfference between dstance and dsplacement. 2. Dscuss the dfference between speed and velocty. 3. Dscuss the dfference between average speed and nstantaneous speed. t 4. Although speed s the magntude of the nstantaneous velocty, s the average speed equal to the magntude of the average velocty? 5. Why can the knematc equatons be used only for moton at constant acceleraton? 6. In dealng wth average veloctes dscuss the statement, "Straght lne moton at 60 mph for hr followed by moton n the same drecton at 30 mph for 2 hr does not gve an average of 45 mph but rather 40 mph." Chapter 3 Knematcs The Study of Moton 7. What effect would ar resstance have on the velocty of a body that s dropped near the surface of the earth? 8. What s the acceleraton of a projectle when ts nstantaneous vertcal velocty s zero at the top of ts trajectory? 9. Can an object have zero velocty at the same tme that t has an acceleraton? Explan and gve some examples. 0. Can the velocty of an object be n a dfferent drecton than the acceleraton? Gve some examples.. Can you devse a means of usng two clocks to measure your reacton tme? tl2. A person on a movng tran throws a ball straght upward. Descrbe the moton as seen by a person on the tran and by a person on the staton platform. 3. You are n free fall, and you let go of your watch. What s the relatve velocty of the watch wth respect to you? tl4. What knd of moton s ndcated by a graph of dsplacement versus tme, f the slope of the curve s (a) horzontal, (b) slopng upward to the rght, and (c) slopng downward? tl5. What knd of moton s ndcated by a graph of velocty versus tme, f the slope of the curve s (a) horzontal, (b) slopng upward at a constant value, (c) slopng upward at a changng rate, (d) slopng downward at a constant value, and (e) slopng downward at a changng rate? 73
Hnts for Problem Solvng To be successful n a physcs course t s necessary to be able to solve problems. The followng procedure should prove helpful n solvng the physcs problems assgned. Frst, as a prelmnary step, read the approprate topc n the textbook. Do not attempt to solve the problems before dong ths. Look at the approprate llustratve problems to see how they are solved. Wth ths background, now read the assgned problem. Now contnue wth the followng procedure.. Draw a small pcture showng the detals of the problem. Ths s very useful so that you do not lose sght of the problem that you are tryng to solve. 2. Lst all the nformaton that you are gven. 3. Lst all the answers you are expected to fnd. 4. From the summary of mportant equatons or the text proper, lst the equatons that are approprate to ths topc. 5. Pck the equaton that relates the varables that you are gven. 6. Place a check mark (^) over each varable that s gven and a queston mark (?) over each varable that you are lookng for. 7. Solve the equaton for the unknown varable. 8. When the answer s obtaned check to see f the answer s reasonable. Let us apply ths technque to the followng example. A car s travelng at 30 ft/s when t starts to accelerate at 0 ft/s2. Fnd (a) the velocty and (b) the dsplacement of the car at the end of 5 s. Problems for Chapter 3. Draw a pcture of the problem. 2. Gven: vj0 = 30 ft/s f=0 t a = 0 ft/s2 r = 5s 3. Fnd: v =? x =? 4. The problem s one n knematcs and the knematc equatons apply. That s, () x = v0t + \at2 (2) v = v0 + at (3) v2 = v20 + 2ax 5. Part a of the problem. To solve for the velocty v, we need an equaton contanng v. Equaton does not contan a velocty term vj, and hence can not be used to solve for the velocty. Equatons 2 and 3, on the other hand, both contan u. Thus, we can use one or possbly both of these equatons to solve for the velocty. 6. Wrte down the equaton and place a check mark over the known terms and a queston mark over the unknown terms:? - j - j - j (2) vj = Uo + at The only unknown n equaton 2 s the velocty v and we can now solve for t. 7. The velocty after 5 s, found from equaton 2 s v = vj0 + at = 30 ft/s + (0ft/s2)(5s) = 30 ft/s + 50 ft/s = 80 ft/s Notce what would happen f we tred to use equaton 3 at ths tme:? J v V? (3) vj2 = v\ + 2ax We can not solve for the velocty v from equaton 3 because there are two unknowns, both v and x. However, f we had solved part b of the problem for x frst, then we could have used ths equaton. 5. Part b of the problem. To solve for the dsplacement x, we need an equaton contanng x. Notce that equaton 2 does not contan x, so we can not use t. Equatons and 3, on the other hand, do contan x, and we can use ether to solve for x. 6. Lookng at equaton, we have? V V V v V () X = V0t + yflf2 7. Solvng for the only unknown n equaton, x, we get x = vv + \at2 = (30 ft/s)(5 s) + 4-(0ft/s2)(5s)2 = 50 ft + 25 ft = 275 ft Note that at ths pont we could also have used equaton 3 to determne x, because we already found the velocty v n part a of the problem. 3. Introducton I. A drver travels 300 m n 5 hr and 25 mn. What s hs average speed n (a) mph, (b) ft/s, (c) km/hr, and (d) m/s? 2. A car travels at 40 mph for 2 hr and 60 mph for 3 hr. What s ts average speed? 3. A man hears the sound of thunder 5 s after he sees the lghtnng flash. If the speed of sound n ar s 343 m/s, how far away s the lghtnng? Assume that the speed of lght s so large that the lghtnng was seen essentally at the same tme that t was created. 4. The earth-moon dstance s 3.84 X 08 m. If t takes 3 days to get to the moon, what s the average speed? 5. Electronc transmsson s broadcast at the speed of lght, whch s 3.00 X 08 m/s. How long would t take for a rado transmsson from earth to an astronaut orbtng the planet Mars? Assume that at the tme of transmsson the dstance from earth to Mars s 7.80 X 07 km. 6. In the game of baseball, some excellent fast-ball ptchers have managed to ptch a ball at approxmately 00 mph. If the ptcher's mound s 60.5 ft from home plate, how long does t take the ball to get to home plate? If the ptcher then throws a change-of-pace ball (a slow ball) at 60 mph, how long wll t now take the ball to get to the plate? 7. Two students are havng a race on a crcular track. Student s on the nsde track, whch has a radus of curvature rx = 250 m, and s movng at the speed O = 4.50 m/s. Wth what speed must student 2 run to keep up wth student f student 2 s on the outsde track of radus of curvature r2 = 255 m? 74 Mechancs
30 - E, 25 c U) F 20 C!) ra 5 n Q 0 A plot of the dsplacement of a car as a functon of tme s shown n the dagram. Fnd the velocty of the car along the paths (a) O-A, (b) A-B, (c) B-C. and (d) C-D. - - t - - H -!- 4 - H- ' I t - t - - - - (- - y lcj - I - B/ - - - / r - I - - - - - - 0 2 3 4 5 6 Tme (s) - V l~ - - /9...A plot of the velocty of a car as a ^-" functon of tme s shown n the >- 5. An arplane travels 000 ft at a constant acceleraton whle takng off. If t starts from rest, and takes off n 25.0 s, what s ts takeoff velocty? n arplane travels 450 km at a constant acceleraton whle takng off". If t starts from rest, and takes off n 30.0 s, what s ts takeoff velocty? 7. A car starts from rest and acqures a velocty of 20.0 mph n 0.0 s. Where s the car located and what s ts velocty at 0.0, 5.0, 20.0, and 25.0 s? 8. A jet arplane goes from rest to a velocty of 250 ft/s n a dstance of 400 ft. What s the arplane's average acceleraton n ft/s2? dagram. Fnd the acceleraton of the car along the paths (a) O-A, (b) A-B, (c) B-C, and (d) C-D. 9. An electron n a vacuum tube acqures a velocty of 5.3 X 08 v,, cm/s n a dstance of 0.25 cm. Fnd the acceleraton of the electron. 2 - - - - 20. A drver travelng at 60.0 mph tres 0 - - h- - f - - - ljc_l - - to stop the car and fnds that the 8. - h - \- brakes have faled. The emergency - - _,_.. A 6 &/' Y '> brake s then pulled and the car comes to a stop n 456 ft. Fnd the o car's deceleraton. CD g \ l \ > 2. An arplane has a touchdown velocty 4 I V of 75.0 knots and comes to rest n 2 - - - - 400 ft. What s the arplane's»- average deceleraton? How long does 0 2 3 4 5 6 7 8 t take the plane to stop? 22. A ptcher gves a baseball a Tme (s) horzontal velocty of 0 ft/s by 0. If an arplane s travelng at 0 movng hs arm through a dstance of knots, what s ts velocty n (a) mph, approxmately 3.00 ft. What s the (b) km/hr, (c) ft/s, and (d) m/s? A average acceleraton of the ball knot s a nautcal mle per hour, and durng ths throwng process? 23. The speedometer of a car reads 60.0 a nautcal mle s equal to 6076 ft. mph when the brakes are appled. The car comes to rest n 4.55 s. How 3.6 The Knematc Equatons far does the car travel before comng n One Dmenson to rest? ll. A grl who s ntally runnng at.00 t24. A body wth unknown ntal velocty moves wth constant acceleraton. At m/s ncreases her velocty to 2.50 the end of 8.00 s, t s movng at a m/s n 5.00 s. Fnd her acceleraton. 2. A car s travelng at 95.0 km/hr. The velocty of 50.0 m/s and t s 200 m from where t started. Fnd the drver steps on the brakes and the car comes to a stop n 60.0 m. What s body's acceleraton and ts ntal the car's deceleraton? velocty. t25. A drver travelng at 25.0 mph sees A tran accelerates from an ntal the lght turn red at the ntersecton. velocty of 20.0 mph to a fnal If her reacton tme s 0.600 s, and velocty of 35.0 mph n.8 s. Fnd the car can decelerate at 8.0 ft/s2, ts acceleraton and the dstance the fnd the stoppng dstance of the car. tran travels durng ths tme. What would the stoppng dstance be 4 ja'tran accelerates from an ntal f the car were movng at 50.0 mph? velocty of 25.0 km/hr to a fnal 26. A drver travelng at 30.0 km/hr sees velocty of 65.0 km/hr n 8.50 s. Fnd the lght turn red at the ntersecton. ts acceleraton and the dstance the If hs reacton tme s 0.600 s, and tran travels durng ths tme. the car can decelerate at 4.50 m/s2, Chapter 3 Knematcs The Study of Moton fnd the stoppng dstance of the car. What would the stoppng dstance be f the car were movng at 90.0 km/hr? t27. A unformly acceleratng tran passes a green lght sgnal at 25.0 km/hr. It passes a second lght 25 m farther down the track, 2.0 s later. What s the tran's acceleraton? What s the tran's velocty at the second lght? x= 25 m 28. A car accelerates from 50.0 mph to 80.0 mph n 26.9 s. Fnd ts acceleraton and the dstance the car travels n ths tme. t29. A motorcycle starts from rest and accelerates at 4.00 m/s2 for 5.00 s. It then moves at constant velocty for 25.0 s, and then decelerates at 2.00 m/s2 untl t stops. Fnd the total dstance that the motorcycle has moved. t30. A car starts from rest and accelerates at a constant rate of 3.00 m/s2 untl t s movng at 8.0 m/s. The car then decreases ts acceleraton to 0.500 m/s2 and contnues movng for an addtonal dstance of 250 m. Fnd the total tme taken. 3.7 The Freely Fallng Body 3. A passenger, n abandonng a snkng ' shp, steps over the sde. The deck s 5.0 m above the water surface. Wth what velocty does the passenger ht the water? 32. How long does t take for a stone to fall from a brdge to the water 30.0 m below? Wth what velocty does the stone ht the water? 33. An automoble travelng at 60.0 mph hts a stone wall. From what heght would the car have to fall to acqure the same velocty? 34. A rock s dropped from the top of a buldng and hts the ground 8.00 s later. How hgh s the buldng? 35. A stone s dropped from a brdge 00 ft hgh. How long wll t take for the stone to ht the water below? 36. A ball s dropped from a buldng 50.0 meters hgh. How long wll t take the ball to ht the ground below? t37. A grl s standng n an elevator that s movng upward at a velocty of 2.0 ft/s when she drops her handbag. If she was orgnally holdng the bag at a heght of 4.00 ft above the elevator floor, how long wll t take the bag to ht the floor? 75
3.9 Projectle Moton n One Dmenson 38. A ball s thrown vertcally upward wth an ntal velocty of 30 ft/s. Fnd ts poston and velocty at the end of 2, 4, 6, and 8 s and sketch these postons and veloctes on a pece of graph paper. 39. A projectle s fred vertcally upward wth an ntal velocty of 40.0 m/s. Fnd the poston and velocty of the projectle at,3, 5, and 7 s. t40. A ball s thrown vertcally upward from the top of a buldng 40.0 m hgh wth an ntal velocty of 25.0 m/s. What s the total tme that the ball s n the ar? 4. A stone s thrown vertcally upward from a brdge 00 ft hgh at an ntal velocty of 50.0 ft/s. How long wll t take for the stone to ht the water below? t42. A stone s thrown vertcally downward from a brdge 00 ft hgh at an ntal velocty of 50.0 ft/s. How long wll t take for the stone to ht the water below? 43. A rock s thrown vertcally downward from a buldng 40.0 m hgh at an ntal velocty of 5.0 m/s. (a) What s the rock's velocty as t strkes the (b) How long does t take for the rock to ht the 44. A baseball batter fouls a ball vertcally upward. The ball s caught rght behnd home plate at the same heght that t was ht. How long was the baseball n flght f t rose a dstance of 00 ft? What was the ntal velocty of the baseball? 3. Projectle Moton n Two Dmensons 45. A projectle s thrown from the top of a buldng wth a horzontal velocty of 5.0 m/s. The projectle lands on the street 85.0 m from the base of the buldng. How hgh s the buldng? 46. To fnd the velocty of water ssung from the nozzle of a garden hose, the nozzle s held horzontally and the stream s drected aganst a vertcal wall. If the wall s 7.00 m from the 76 nozzle and the water strkes the wall 0.650 m below the horzontal, what s the velocty of the water? 47. A bomb s dropped from an arplane n level flght at a velocty of 970 km/hr. The alttude of the arcraft s 2000 m. At what horzontal dstance from the ntal poston of the arcraft wll the bomb land? t48. A cannon s placed on a hll 20.0 m above level ground. A shell s fred horzontally at a muzzle velocty of 300 m/s. At what horzontal dstance from the cannon wll the shell land? How long wll ths take? What wll be the shell's velocty as t strkes ts target? 49. A shell s fred from a cannon at a velocty of 300 m/s to ht a target 3000 m away. At what angle above the horzontal should the cannon be amed? 50. In order to ht a target, a marksman fnds he must am 0.0 cm above the target, whch s 300 m away. What s the ntal speed of the bullet? 5. A golf ball s ht wth an ntal velocty of 75 ft/s at an angle of 50.0 above the horzontal, (a) How hgh wll the ball go? (b) What s the total tme the ball s n the ar? (c) How far wll the ball travel horzontally before t hts the 52. A projectle s thrown from the ground wth an ntal velocty of 20.0 m/s at an angle of 40.0 above the horzontal. Fnd (a) the projectle's maxmum heght, (b) the tme requred to reach ts maxmum heght, (c) ts velocty at the top of the trajectory, (d) the range of the projectle, and (e) the total tme of flght. Addtonal Problems 53. A mssle has a velocty of 0,000 mph at "burn-out," whch occurs 2 mn after gnton. Fnd the average acceleraton n (a) ft/s2, (b) m/s2, and (c) n terms of g, the acceleraton due to gravty at the surface of the earth. 54. A block sldes down a smooth nclned plane that makes an angle of 25.0 wth the horzontal. Fnd the acceleraton of the block. If the plane s 0.0 meters long and the block starts from rest, what s ts velocty at the bottom of the plane? How long does t take for the block to get to the bottom? t55. At the nstant that the traffc lght turns green, a car startng from rest wth an acceleraton of 7.00 ft/s2 s passed by a truck movng at a constant velocty of 30.0 mph. (a) How long wll t take for the car to overtake the truck? (b) How far from the startng pont wll the car overtake the truck? (c) At what velocty wll the car be movng when t overtakes the truck? 56. At the nstant that the traffc lght turns green, a car startng from rest wth an acceleraton of 2.50 m/s2 s passed by a truck movng at a constant velocty of 60.0 km/hr. (a) How long wll t take for the car to overtake the truck? (b) How far from the startng pont wll the car overtake the truck? (c) At what velocty wll the car be movng when t overtakes the truck? t57. A boat passes a buoy whle movng to the rght at a velocty of 8.00 m/s. The boat has a constant acceleraton to the left, and 0.0 s later the boat s found to be movng at a velocty of 3.00 m/s. Fnd (a) the acceleraton of the boat, (b) the dstance from the buoy when the boat reversed drecton, (c) the tme for the boat to return to the buoy, and (d) the velocty of the boat when t returns to the buoy. t58. Two trans are ntally at rest on parallel tracks wth tran 50.0 m ahead of tran 2. Both trans accelerate smultaneously, tran at the rate of 2.00 m/s2 and tran 2 at the rate of 2.50 m/s2. How long wll t take tran 2 to overtake tran? How far wll tran 2 travel before t overtakes tran? t59. Repeat problem 58 but wth tran ntally movng at 5.00 m/s and tran 2 ntally movng at 7.00 m/s. t60. A polceman drvng at 55.0 mph observes a car 200 ft ahead of hm speedng at 80.0 mph. If the county lne s 200 ft away from the polce car, what must the acceleraton of the polce car be, n order to catch the speeder before he leaves the county? 6. A polcewoman drvng at 80.0 km/hr observes a car 50.0 m ahead of her speedng at 20 km/hr. If the county lne s 400 m away from the polce car, what must the acceleraton of the polce car be n order to catch the speeder before he leaves the county? t62. Two trans are approachng each other along a straght and level track. The frst tran s headng east at 70.0 mph, whle the second tran s headng west at 45.0 mph. When they are.50 mles apart they see each other and start to decelerate. Tran decelerates at 5.00 ft/s2, whle tran 2 decelerates at 3.00 ft/s2. Wll the trans be able to stop or wll there be a collson? Mechancs
63. Two trans are approachng each other along a straght and level track. The frst tran s headng south at 25 km/hr, whle the second tran s headng north at 80.0 km/hr. When they are 2.00 km apart, they see each other and start to decelerate. Tran decelerates at 2.00 m/s2, whle tran 2 decelerates at.50 m/s2. Wll the trans be able to stop or wll there be a collson? t64. A boy n an elevator, whch s descendng at the constant velocty of 5.00 m/s, jumps to a heght of 0.500 m above the elevator floor. How far wll the elevator descend before the boy returns to the elevator floor? 65. The acceleraton due to gravty on the moon s.62 m/s2. If an astronaut on the moon throws a ball straght upward, wth an ntal velocty of 25.0 m/s, how hgh wll the ball rse? t66. A helcopter, at an alttude of 300 m, s rsng vertcally at 20.0 m/s when a wheel falls off. How hgh wll the wheel go wth respect to the How long wll t take for the wheel to ht the ground below? At what velocty wll the wheel ht the 300 m t67. A ball s dropped from the roof of a buldng 40.0 m hgh. Smultaneously, another ball s thrown upward from the ground and colldes wth the frst ball at half the dstance to the roof. What was the ntal velocty of the ball that was thrown upward? t68. A ball s dropped from the top of a 40.0-m hgh buldng. At what ntal velocty must a second ball be thrown from the top of the buldng 2.00 s later, such that both balls arrve at the ground at the same tme? t69. Show that the range of a projectle s the same for ether a projecton angle of 45.0 +8 or an angle of 45.0-8. 70. A projectle hts a target.50 km away 0.5 s after t was fred. Fnd (a) the elevaton angle of the gun and (b) the ntal velocty of the projectle. 7. A football s kcked wth an ntal velocty of 70.0 ft/s at an angle of 65.0 above the horzontal. Fnd (a) how long the ball s n the ar, (b) how far down feld the ball lands, (c) how hgh the ball rses, and (d) the velocty of the ball when t strkes the ground. f72. A baseball s ht at an ntal velocty of 0 ft/s at an angle of 45.0 above the horzontal. Wll the ball clear a 0.0 ft fence 300 ft from home plate for a home run? If so, by how much wll t clear the fence? t73. A ball s thrown from a brdge 00 m hgh at an ntal velocty of 30.0 m/s at an angle of 50.0 above the horzontal. Fnd (a) how hgh the ball goes, (b) the total tme the ball s n the ar, (c) the maxmum horzontal dstance that the ball travels, and (d) the velocty of the ball as t strkes the ground. 74. A ball s thrown at an angle of 35.5 below the horzontal at a speed of 22.5 m/s from a buldng 20.0 m hgh, (a) How long wll t take for the ball to ht the ground below? (b) How far from the buldng wll the ball land? f75. Usng the knematc equatons for the x- and ^-components of the dsplacement, fnd the equaton of the trajectory for two-dmensonal projectle moton. Compare ths equaton wth the equaton for a parabola expressed n ts standard form. t76. Usng the knematc equatons, prove that f two balls are released smultaneously from a table, one wth zero velocty and the other wth a horzontal velocty o0r, they wll both reach the ground at the same tme. Interactve Tutorals y 77. A tran accelerates from an ntal velocty of 20.0 m/s to a fnal velocty of 35.0 m/s n.8 s. Fnd ts acceleraton and the dstance the tran travels n ths tme. y 78. A ball s dropped from a buldng 50.0 m hgh. How long wll t take the ball to ht the ground below and wth what fnal velocty? y 79. A golf ball s ht wth an ntal velocty u = 53.0 m/s at an angle 8 = 50.0 above the horzontal. (a) How hgh wll the ball go? (b) What s the total tme the ball s n the ar? (c) How far wll the ball travel horzontally before t hts the y80. Instantaneous velocty. If the equaton for the dsplacement x of a body s known, the average velocty throughout an nterval can be computed by the formula 0 g = (Ax) I (At) The nstantaneous velocty s defned as the lmt of the average velocty as At approaches zero. That s, v = lm (Ax)/(At) At -.0 For an acceleraton wth a dsplacement gven by x = 0.5 at2, use dfferent values of At to see how the average velocty approaches the nstantaneous velocty. Compare ths to the velocty determned by the equaton v = at, and determne the percentage error. Plot the average velocty, (Ax)/(At), versus At. y 8. Free-fall and generalzed onedmensonal projectle moton. A projectle s fred from a heght y0 above the ground wth an ntal velocty u0 n a vertcal drecton. Fnd (a) the tme t, for the projectle to rse to ts maxmum heght, (b) the total tme t, the ball s n the ar, (c) the maxmum heght ym x of the projectle, (d) the velocty ue of the projectle as t strkes the ground, and (e) the locaton and velocty of the projectle at any tme /. (f) Plot a pcture of the moton as a functon of tme. y82. Generalzed two-dmensonal projectle moton. A projectle s fred from a heght y0 above the horzontal wth an ntal velocty u0 at an angle 8. Fnd (a) the tme tr for the projectle to rse to ts maxmum heght; (b) the total tme t, the ball s n the ar; (c) the maxmum dstance the ball travels n the x-drecton, xmax before t hts the ground; (d) the maxmum heght ym3lx of the projectle; (e) the velocty ug of the projectle as t strkes the ground; and (f) the locaton and velocty of the projectle at any tme t. (g) Plot a pcture of the trajectory. Chapter 3 Knematcs The Study of Moton 77