12.1 A rocket is launched vertically at time t = 0. The elevation of the rocket is given byy = 0.13t4 + 4.1t3 + 0.1t m!here tis in seconds. "etermine the ma#imum velocity of the rocket and theelevation at !hich itoccurs.f V g xo+ =

112.2 $hen an ob%ect is tossed vertically u&!ard on the surface of a &lanet' the ensuing motion in theabsence of atmos&heric resistance can be described by!here g and v0 are constants. (a) "erive the e#&ressions for the velocity and acceleration of the ob%ect.*se the results to sho! that v0 is the initial s&eed of the body and that g re&resents the gravitationalacceleration. (b) "erive the ma#imumheight reached by the ob%ect and the total time of flight. (c)+valuate the results of ,art (b) for v0 = -0 km.h and g = -./m.s (surface of the earth).12.3 The &osition of a &article moving along the x0a#is is described by x = t 3 10/t m !here t is the time in seconds. 1or the time interval t = 0 to t = 10 s' (a) &lot the &osition' velocity' andaccelerationasfunctionsoftime2 (b)findthedis&lacementofthe&article2 and(c)determinethedistance traveled by the &article.12.4 The &osition of a &article that moves along the x0a#is is given byx = t3 3t 43t m!here t is the time in seconds. "etermine the &osition' velocity' acceleration' and distance traveled at t= / s.mtt x-03

=12.5 The &osition of a car moving on a straight high!ay is given by!here t is the time in seconds. "etermine (a) the distance traveled by the car before it comes to a sto&2 and (b) the ma#imum velocity reached by the car.12.6 A body is released from rest at A and allo!ed to fall freely. 4ncluding the effects of air resistance' the &osition of the body as a function of the ela&sed time is!here v0 and t0 are constants. (a)"erive the e#&ression for the s&eed v of the body. *se the result to e#&lain !hy v0 is called the terminal velocity. (b) "erive the e#&ressions for the acceleration a of the body as a function of t and asafunction of v.) (.ot to o oe t t t v x+ =12.7 A bead moves along a straight 500mm !ire that lies along the x0a#is. The &osition of the bead isgiven byx = t 10t mm!here x is measured from the center of the !ire' and t is the time in seconds. "etermine (a) the time !hen the bead leaves the !ire2 and (b) the distance traveled by the bead from t = 0 until it leaves the !ire.12.8 A &article moves along the curve x = 1y' !here x and y are measured in millimeters. The x0coordinate varies !ith time according tox = 4t mm!here the time t is in seconds. "etermine the magnitudes of the velocity and acceleration vectors !hen t =s.) cos

11 ( t R x + =12.9 The circular cam of radius R and eccentricity R. rotates clock!ise !ith a constant angular s&eed . The resulting vertical motion of the flat follo!er A can be sho!n to be(a) 6btain the velocity and acceleration of the follo!er as a function of t. (b) 4f !ere doubled' ho! !ould the ma#imum velocity and ma#imum acceleration of the follo!er be changed712.10 The elevator A is lo!ered by a cable that runs over &ulley B. 4f the cable un!inds from the !inch C at the constant s&eed v0' the motion of the elevator is"etermine the velocity and accelerationof the elevator in terms of the time t. ) ( b b t v xo =12.11 A missile is launched from the surface of a &lanet !ith the s&eed v0 at t = 0. According to the theory of universal gravitation' the s&eed v of the missile after launch is given by

!here g is the gravitational acceleration on the surface of the &lanet and r0 is the mean radius of the &lanet. (a) "etermine the acceleration of the missile in terms of r. (b) 1ind the escape velocity, that is' the minimum value of v0 for !hich the missile !ill not return to the &lanet. (c) *sing the result of ,art(b)' calculate the esca&e velocity for earth' !here g = -./ m.s and r0 = 5400 km12.12 The coordinates of a &article undergoing &lane motion arex = 13 t m y = 13 10t + t m!here t is the time in seconds. 1ind the velocity and acceleration vectors at (a) t = 0 s2 and (b) t = 3 s 12.13 A &ro%ectile fired at O follo!s a &arabolic tra%ectory' given in &arametric form byx = 55t y= /5t 4.-1t!here x and y are measured in meters and t in seconds. "etermine (a) the acceleration vector throughout the flight2 (b) the velocity vector at O2 (c) the ma#imum height h2 and(d) the range L.12.14 An automobile goes do!n a hill that has the &arabolic cross sectionsho!n. Assuming that the hori8ontal com&onent of the velocity vector has aconstant magnitude v0' determine (a) the e#&ression for the s&eed of theautomobile in terms of x2 and (b) the magnitude and direction of the acceleration.12.15 The &osition of a &article in &lane motion is defined byx = a cos t y= b sin t!here a >b' and is a constant. (a) 9ho! that the &ath of the &article is an elli&se. (b) ,rove that the acceleration vector is al!ays directed to!ard the center of the elli&se.12.16 $hen a taut string is un!ound from a stationary cylinder' the end B of the stringgenerates a curve kno!n as the involte of a circle. 4f the string is un!ound at the constantangular s&eed ' the e:uation of the involute isx = R cos t + Rt sin t y= R sin t Rt cos t!here R is the radius of the cylinder. 1ind the s&eed of B as a function of time. 9ho! that thevelocity vector is al!ays &er&endicular to the string12.17 $hen a !heel of radius R rolls !ith a constant angular velocity ' the&oint B on the circumference of the !heel traces out a curve kno!n as a cycloi!,the e:uation of !hich isx = R"t sin t# y = R"1 cos t#(a) 9ho! that the velocity vector of B is al!ays &er&endicular to BC. (b) 9ho! that the acceleration vector of B is directed along B$.12.89 The volleyball player serves the ball at point A with the speed vo at the angle =70o. What is the largest vo for which the ball will not hit the ceiling. 12.90 A proectile is la!nched at A with the velocity vo= 20"#s at the angle = $%0. &indthe height h of the i"pact point ' on the vertical wall. (eglect air resistance.12.91 A "issile is la!nched hori)ontally at A with the speed vo= 200"#s. *nowing thatthe range of the "issile is +=1,00"- calc!late the la!nch height h and the ti"e of flight.12.92 A proectile is la!nched hori)ontally at A with the speed of vo. The ti"e of flight is10s- and the path of the proectile at ' is inclined at 20o with the hori)ontal. .eter"inevo- the range +- and the la!nch height h. !se the /0 c!sto"ary !nits. 12.91 A proectileisfired hori)ontally at100"#sdown theinclinedplane..rawtheacceleration- velocity-andthepositiondiagra"s. /sethediagra"todeter"inethe"a2i"!" height h perpendic!lar to the plane- the range + along the plane and the ti"eof flight. (eglect air resistance.12.9, A proectile is la!nched at an elevated target with initial speed vo= 220"#s in thedirection shown. .eter"ine the ti"e of flight and the range +.12.9% A car is initially at rest accelerates along a straight- level road according to thediagra" shown. .eter"ine 3a4 the "a2i"!" speed5 and 3b4 the distance traveled by carwhen the "a2i"!" speed is reached. 12.9$ As!bwaytrainstops at twostations that are26"apart. The"a2i"!"acceleration and deceleration of the train are $.$"#s2 and %.%"#s2- respectively- and the"a2i"!" allowable speed is 906"#h. find the shortest possible ti"e of travel betweenthe two stations. 12.97 A train is bro!ght to an e"ergency stop in 1$ seconds- the deceleration beingas shown in the diagra"- co"p!te the speed of the train before the bra6es were appliedand the stopping distance. 12.98 An airplane lands on a levelr!nway at the speed of ,0"#s. for the first threeseconds after to!chdown- the reverse thr!st of the propellers ca!ses a deceleration of1.2"#s2. &or the ne2t five seconds- the wheel bra6es are applied prod!cing anadditionaldeceleration of 1.8 "#s2. Then the reverse thr!sters are sh!tdown- and theplane is bro!ght to a stop with only the wheel bra6es. .raw the acceleration- velocity-and position diagra"s. 7ow far does the airplane travel on the r!nway before it co"esto a stop8