KINEMATICS ASSIGNMENT -I

1. A train starts from a station with a constant acceleration of 0.40 2sm . A passenger arrives at the

station 6.0s after the end of the train left the very same point. What is the least constant speed at whichshe can run and catch the train ?

2. Ball A is dropped from the top of a building at the same instant when ball B is thrown vertically upwardfrom the ground. When the balls collide, they were moving in opposite directions and the speed of Ais twice the speed of B. At what fraction of the height of the building did the collision occur?

3. A baseball is struck by a bat, and 3s later it is caught 30m away.(a) If it was 1m above the ground when struck and caught, what was the greatest height it reachedabove the ground ?(b) What were its horizontal and vertical components of velocity when it was struck ?(c) What was its speed when it was caught ?(d) At what angle with the horizontal did it leave the bat ? (Neglect air resistance.)

4. A boy stands 4m from a vertical wall and throws a ball. The ball leaves the boy’s hand at 2m above theground with initial velocity v =10i +10j m/s. When the ball hits the wall, its horizontal component ofvelocity is reversed and its vertical component remains unchanged. Where does the ball hit the ground?

5. An elevator without a ceiling is ascending with a constant speed of 10m/s. A boy in the elevator throwsa ball vertically upward, from a height 2m above the elevator floor, just as the elevator floor is 28 mabove the ground. The initial speed of the ball with respect to the elevator is 20m/s.(a) What is the maximum height attained by the ball ? (b) How long does it take for the ball to return tothe elevator floor ?

6. A body A begins to move with the initial velocity v=2m/s and continues to move at constant accelera-tion a. In ∆ t=10 sec, after the body A begins to move, a body B departs from the same point with aninitial velocity v=12m/s and moves with the same acceleration a. What is the maximum acceleration aat which the body can overtake the body A ?

7. A motorboat going downstream overcame a raft at a point A; 60 minutes later it turned back and aftersome time passed the raft at a distance 6 km from the point A. Find the flow velocity assuming the dutyof the engine to be constant.

8. At the moment t=0 a particle leaves the origin and moves along the positive X-axis. Its velocity

varies with time as v=v0 (1-t/T) where v

0 is the initial velocity vector with magnitude 10 cm/s; T = 5 sec. Find:

(a) the x coordinate of the particle at t=6 sec. t=10 sec. and t=20 sec;(b) the moments of time when the particle is at the distance 10cm from the origin;(c) the distances covered by the particle during the first 4 and 8 sec.

9. A motorboat has two throttle positions on its engine. The high speed position propels the boat at 10km hr–1 in still water and the low position gives half the higher speed. The boat travels from its dockdown-stream on a river with the throttle at high position. The return trip took 15% longer time than itdid for the down-stream trip. Find the value of the water current in the river.

10. A train starting from rest has to run 1 km to acquire its full speed of 60 km hr–1 and half a km to cometo rest from the full speed, under the action of its brakes. If some where mid-way between twostations, 1 km of the railway track is under repair and the speed over this span is limited to 20 km hr–

1 , find how late (minimum value) the train will be when it reaches the other station, assuming the trainotherwise runs at full speed.

11. An elevator car whose floor-ceiling distance is equal to 2.7 m starts ascending with constant accelera-tion 1.2 m/s2; 2.0 s after the start, a bolt begins falling from the ceiling of the car.Find:(a) the bolt’s free fall time,(b) the displacement and the distance covered by the bolt during the free fall in the referenceframe fixed to the elevator shaft.

12. A boy standing on a long railroad car throws a ball straight upwards. The car is moving on the

horizontal road with an acceleration of 1

and the projection velocity in the vertical direction is9.8 m/s. How far behind the boy will the ball fall on the car ?

13. An aeroplane has to go from a point A to another point B, 500km away due 30° east of north. A windis blowing due north at a speed of 20 m/s. The air-speed of the plane is 150 m/s. (a) Find thedirection in which the pilot should head the plane to reach the point B . (b) Find the time taken by theplane to go from A to B.

14. You throw a ball with a speed of 25m/s at an angle of 40o above the horizontal directly towards a wall.The wall is 22 m from the release point of the wall. a) How long is the ball in the air before it hits the wall? b) How far above the release point does the ball hit the wall? c) What are the horizontal and verticalcomponents of its velocity as it hits the wall ? d) Has it passed the highest point on its trajectory whenit hits ?

15. A ball rolls horizontally off the top of a staircase with a speed of 5ft/s. The steps are 8in. high and 8 in.wide. Which step will the ball hit first ?

16. A particle is projected with a velocity 2 (ag)1/2 so that it clears two walls of equal height a, which are adistance 2a apart from each other. Prove that the time of passing between the walls is 2(a/g)1/2 .

17. The kicker on a football team can give the ball an initial speed of 25 m/s. Within what two elevationangles must he kick the ball if he is to score a goal from a point 50m in front of the goalpost whosehorizontal bar is 3.44m above the ground? (You might want to usesin2A+cos2A=1 to get a relation between tan2A and 1/cos2A and then solve the resulting quadraticequation)

18. A rocket is launched from rest and moves in a straight line at 70.0o above the horizontal with anacceleration of 46.0 m/sec . After 30.0s of the linear powered flight, the engine is shut off and therocket follows a parabolic path back to the earth. Assume that he free fall acceleration is 9.8 m\s2

throughout and that effects of the air can be ignored.(a) Find the time of the flight from the launch to the impact? (b) What is the maximum altitudereached? (c) What is the distance from the launched to the impact point?

19. A stone is thrown in such a manner that it would just hit a bird at the top of a tree and afterwardsreach a height double that of the tree. If at the moment of throwing the stone the bird fly awayhorizontally, show that notwithstanding this, the stone will hit the bird if the ratio of its horizontalvelocity to that of the bird is( 21/2 +1) / 2 .

20. A particle is projected up an inclined plane of inclination β at an elevation α to the horizon. Showthat :

a) tan α = cot β + 2 tan β , if the particle strikes the plane at right angles.

b) tan α = 2 tan β , if the particle strikes the plane horizontally.

21. A gun ,kept on a horizontal road, is used to hit a car travelling along the same road away from the gunwith a uniform speed of 72 km/h. The car is at a distance of 500 m from the gun, when the gun is firedat an angle of 45° with the horizontal. Find a) the distance of the car from the gun when the shell hits itb) the speed of projection of the shell from the gun.

22. A particle is projected with speed u so as to strike at right angles on a plane through the point ofprojection inclined at 30° to the horizontal. Find the range R on this inclined plane.

23. A wooden boxcar is moving along a straight railroad track at speed v1. A sniper fires a bullet (initial speed

v2 ) at it from a high-powered rifle. The bullet passes through both walls of the car, its entrance and exit

holes being exactly opposite to each other as viewed from within the car. From what direction , relativeto the track, was the bullet fired ? Assume that the bullet did not deflect upon entering the car, but that itsspeed decreased by 20 %. Take v

1 = 85 km/h and v

2 = 650 m/s. ( Why don’t you need to know the

width of the car ?).

24. A passenger car is driving over a level highway behind a truck. A stone got struck between the doubletyres of the rear wheels of the truck . At what distance should the car follow the truck so that the stonewill not strike it if it flies out from between the tyres ? Both vehicles have a speed of 50 km/h.

25. A particle moves in the XY plane with constant acceleration w directed along the negative Y-axis. Theequation of motion of the particle has the form y = ax - bx2 , where a and b are positive constants. Findthe velocity of the particle at the origin of the coordinates.

26. A particle moves in the XY plane with velocity v = a i + bx j where i and j are the unit vectors of the Xand Y axes , and a and b are constants. At t = 0, the particle is at the origin. Find the equation of theparticle’ s trajectory y (x).

27. A person travelling eastwards finds that the wind appears to blow directly from north. On doubling hisspeed, it seems to come from north-east. If he trebles his speed, in what direction it would appear toblow ?

28. A boat is approaching the shore with a speed of 53 m/s. At the instant when it is at adistance of 303m from the shore, a stone is to be projected at an elevation of 30o for it to just reach the shore. Whatshould be the speed of the stone relative to the boat ?

29. An elevator is going up with an upward acceleration of 1 m/sec2. At the instant when its velocity is 2 m/s, stone is projected upward from its floor with a speed of 2 m/s relative to the elevator floor, at anelevation of 30o. (a) Calculate the time taken by the stone to return to the floor. (b) Sketch the path of theprojectile as observed by an observer outside the elevator. (c) If the elevator was moving with a down-ward acceleration equal to g, how would the motion be altered. (d) Find, range of the stone over thefloor of the lift. (g = 10 ms-2)

30. Two bodies were thrown simultaneously from the same point : one, straight up, and the other, at an angleof θ =60o to the horizontal. The initial velocity of each body is equal to v

0 = 25 m/s. Neglecting the air

drag, find the distance between the bodies t=1.70 s later.

31. Two particles move in a uniform gravitational field with an acceleration g. At the initial moment theparticles were located at one point and moved with velocities v

1 = 3.0 m/s and v

2 = 4.0 m/s horizon-

tally in opposite directions. Find the distance between the particles at the moment when their velocityvectors become mutually perpendicular.

32. From a point in a smooth horizontal plane, a particle is projected with velocity u at an angle α to thehorizontal. If the coefficient of restitution between the plane and the particle is e, find the distance de-scribed along the plane before the particle ceases to rebound.

33. A gun is fired from a moving platform and the ranges of the shot are observed to be R and S when theplatform is moving forward or backward respectively with velocity V. Prove that the elevation of the gun

is tan [( )

( )]−−−− −−−−

++++1

2

24

g R S

V R S.

34. A basketball player throws the ball with initial velocity v0 at an angle θ with the horizontal to the hoop

which is located at a horizontal distance L and at a height h above the point of release, as shown in figure.

(a) Show that the initial speed required is given by

vgL

h L02

22=

−cos (tan / )θ θ

(b) Show that the angle α to the horizontal at which it reaches the hoop is givenby tan / tanα θ= −2h L

35. Two boats A and B move away from a buoy anchored at the middle of a river along the mutuallyperpendicular straight lines; the boat A moves along the river, and the boat B moves across the river. Having

moved off an equal distance from the buoy, the boats returned. Find the ratio of times of motion of boatstA / t

B if the velocity of each boat with respect to water is β times greater than the stream velocity.

36. An airplane has a speed of 135 km/hr in still air. It is flying straight north so that it is at all times directlyabove the north-south highway. A ground observer tells the pilot by radio that a 70 km/hr wind isblowing, but neglects to tell him the wind direction. The pilot observes that in spite of the wind, he cantravel 135 km along the highway in one hour. In other words, his ground speed is the same as if therewere no wind.

(a) What is the direction of the wind ?(b) What is the heading of the plane, that is, the angle between its axis and the highway ?

Note: Substituting W for E in the above gives another solution.

37. Two shots are fired from a gun at the top of a cliff with the same speed v0, at angles of projection α and

β respectively. If the shots strike the horizontal through the foot of the cliff at the same point, determinethe height of the cliff.

38. Two shells are projected simultaneously from the same point with the same initial velocity so as to movein the same vertical plane, their initial directions of motion being α and α′ respectively with the horizontal.

Prove that the shells move so that the line joining them makes the same constant angle α α+ '

2 with the

vertical.

39. Two bodies are projected from the same point with equal velocities in such directions that they both

strike the same point on a plane whose inclination is β . If α be the angle of projection of the first, show

that the ratio of their times of flight is sin )( β+α : cos α .

40. Two cyclists move towards each other. The first cyclist, whose initial velocity v01

= 5.4 km/h, descendsthe hill, gathering speed with an acceleration a

1 = 0.2 m/s2. The second cyclist, whose initial velocity v

02

= 18 km/h climbs the hill with an acceleration a2 = −0.2 m/s2. How long does it take for the cylist to meet

if the distance x0 separating them at the initial instant of time is 195 m ?

KINEMATICS ASSIGNMENT II

1. The velocity of a particle moving in the positive direction of X-axis varies as v =α (x)1/2 where a is apositive constant. Assuming that at t=0 the particle is at x=0, find:(a) the time dependence of the velocity and the acceleration of the particle;(b) the mean velocity of the particle averaged over the time that the particle takes to coverthe first s meters of the path.

2. A point moves rectilinearly with a deceleration whose modulus depends on the velocity v of the particleas a=α (v)1/2 where α is a positive constant. At t=0 the velocity is v

0. What distance will it traverse

before it stops ? What time will it take to cover that distance ?

3. To a close approximation, the pressure behind a bullet varies inversely with the position x of the bulletalong the barrel. Thus the acceleration of the bullet may be written as a = k/x where k is a constant. Ifthe bullet starts from rest at x = 7.5 mm and if the muzzle velocity of the bullet is 600 m/s at the end of750 mm barrel, compute the acceleration of the bullet as it passes the mid point of the barrel at x = 375mm ?

4. Two trains A and B are approaching each other on straight track, the former with a uniform velocity of25 m/s and the latter with 15 m/s. When they are 225 m apart, brakes are simultaneously applied toboth of them. The deceleration given by the brakes to the train B increases linearly with time by 0.3 m/s2 every second, while the train A is given a uniform deceleration. (A) What must be the minimumdeceleration of A so that the trains do not collide ? (b) What is the time taken by the trains to come tostop ?

5. A particle is thrown up with an initial speed v0 . There is a resisting acceleration –kv due to air, which

is proportional to the instantaneous velocity v of the particle, k being a constant. Show that the timetaken to reach the highest point is given by (1/k) ln ( 1+ v

0k/g).

6. At what angle α above the horizon should a stone be thrown from a steep bank for it to fall into thewater as far as possible from the bank? The height of the bank h = 20m and the initial velocity of thestone v = 14m/s.

7. A man wants to cross a river 500 m wide. His rowing speed (relative to water) is 3000 m/s. The riverflows at a speed of 2000 m/h. The man’s walking speed on the shore is 5000 m/s. a) Find the path(combined rowing and walking) he should take to get to the point directly opposite his starting point inthe shortest time. b) How long does it take ?

8. A battleship steams due east at 24 km/h. A submarine 4 km away fires a torpedo that has a speed of 50km/h. If the bearing of the ship as seen from the submarine is 20o east of north, a) in what directionshould the torpedo be fired to hit the ship, and b) what will be the running time for the torpedo to reachthe battleship ?

9. A weightless inextensible rope rests on a stationary wedge forming an angle α with the horizontal(Figure). One end of the rope is fixed to the wall at point A. A small load is attached to the rope at point

B. The wedge starts moving to the right with a constant acceleration a. Determine the acceleration 1aof the load when it is still on the wedge.

a

α

B

A

10. Two particles are simultaneously thrown from roofs of two high buildings, as shown in fig. Their veloci-

ties of projection are 12 −ms and 114 −ms respectively. Horizontal and vertical separation between

points A and B is 22m and 9m respectively.

A

12 −= msu

°45

°45

114 −= msv m9

m22

B

Calculate minimum separation between the particles in the process of their motion.

11. A ball of mass m is thrown at an angle of °45 to the horizontal from top of a 65m high tower AB as

shown is fig. Another identical ball is thrown with velocity 120 −ms horizontally towards AB from top

of a 30m high tower CD one second after the projection of first ball. Both the balls move in samevertical plane. If they collide in mid air.

A

°45B

1210 −ms

m65

dC

D

m30

120 −ms

i) Calculate distance AC.ii) During collision, the two balls get stuck together, calculate the distance between A andthe point on the ground , at which the combined ball strikes.

Given, 210 −= msg

12. A particle is moving along a vertical circle of radius mR 20= with a constant speed 14.31 −= msv as

shown in fig. Straight line ABC is horizontal and passes through the centre of the circle. A shell is fired

from point A at the instant when particle is at C. If distance AB is m320 and shell collides with the

particle at B, calculate.

AB

Cm20

θ

i) smallest possible value of the angle θ of projectionii) corresponding velocity u of projection

13. A particle is projected from point O on the ground with velocity 155 −= msu at angle ( )5.0tan 1−=α .

It strikes at a point C on a fixed smooth plane AB having inclination of °37 with horizontalIf the particle does not rebound, calculate

m3

10O

°37

155 −msC

B

A

i) co-ordinates of point C in reference to co-ordinate system shown if fig.ii) maximum height from the ground to which the particle rises.

14. Two identical shells are fired from a point on the ground with same muzzle velocity at angles of eleva-

tion °= 45α and 1tan−=β 3 towards top of a cliff, 20m away from point of firing.

If both the shells reach the top simultaneously, calculatei) muzzle velocityii) height of the cliff andiii) time interval between two firingsIf just before striking the top of cliff the two shells get stuck together, considering elastic collision ofcombined body with the top, calculateiv) maximum height reached by the combined body.

15. A shell of mass gmm 700= is fired form ground with a velocity 140 −ms . At highest point of its trajec-

tory, it collides inelastically with a ball of mass kgM 3.1= , suspended by a flexible thread of length1.40m. If thread deviates through an angle of °120 before slackening, calculatei) angle of projection of shell,ii) maximum height of combined body from ground andiii) distance between point of suspension of ball and point of projection of shell.

16. A projectile is fired up an incline with an initial speed v0 at an angle θ to the horizontal, as shown in

figure.

(a) If the angle of the incline is α to the horizontal, find the range along the inclined plane

(b) Find the maximum value of R along the inclined plane

Rv0

θα

17. A shot is fired with a velocity u at a vertical wall whose distance from the point of projection is x. Provethat the greatest height above the level of the point of projection at which the bullet can hit the wall is

u g x

gu

4 2 2

22

−

18. From a point A on a bank of channel with still waters, a man must get to a point B on the opposite side

of channel. The man uses a boat to travel across the channel and then walks on the land to reach pointB. His rowing and walking speeds are v

1 and v

2 respectively. Prove that the fastest way for the man to

get from A to B is to select α1 and α

2 in such a manner that

2

1

2

1

v

v

sin

sin =αα

.

19. A particle moving in a straight line is subject to a resistance which produces the retardation kv3, wherev is the velocity and k is a constant. Show that the velocity v and the time t are given in terms of the

distances by the equations 2ks

2

1

u

st,

kus1

uv +=

+=

21. A cable attached to the car at A passes over the small fixed pulley at B and around the drum at C. If the

vehicle moves with a constant speed ,dt

dxv0 = determine an expression for the acceleration of a point

P on the cable between B and C in terms of θ. Also express d

dt

2

2

θ in terms of θ.

22. A plane contains two straight line intersecting each other at an angle α. The lines start moving perpendicularto themselves in the same plane as shown in figure, with velocities v

1 and v

2 respectively. Show that the

velocity of the point of intersection of the two lines is

v v v v12

22

1 22+ + cos

sin

αα

Assignment I

1. 4,8 m/s 2.3

2

3. (a) 12.0m (b) 10 m/s; 14.7 m/s (c) 17.8m/s (d) 55.8°4. 13.8m bhind the boy.5. (a) 76m above the ground (b) 4.1s6. a=1m/s2. 7. 3 km/hr.8. (a) x = 0.24m, 0,–2m (b) 1.1, 8.9 and 11 sec., (c) 24 and 34 cm respectively.9. 3 km hr–1 10. 160 s11. (a) 0.7 s; (b) 0.7 and 1.3 m respectively 12. 2m

13. (a) 1sin− (1/15) east of the line AB (b) 50 min14. a) 1.15 s b)12 m c) 19.2 m/s, 4.8 m/s d)No.15. The third16. -17. a) Between the angle 31o and 63o above the horizontal.18. a) 310 sb) 105 km c)139 km.19. -20. -21. a) 746.9 m b) 85.56 m/s 22. R = 4u2/7g.23. 87o from the direction of motion of the car. 24. s = 19.6 m

25.