Section : A Definition, Projectile on a horizontal plane

1. The velocity of a body at the highest point of its path is not zero. If so how it is projected ?

2. In order to project a body for maximum range, what is the condition ?

3. What is the angle between the directions of velocity & acceleraiton at the highest point of a projectile path ?

4. At what point of the projectile path the speed is minimum ?

5. Two bodies are projected at angles and (90 – ) to the horizontal with the same speed. Find the ratio of theirtimes of flight?

6. In above question find the ratio of the maximum vertical heights ?

7. In the case of a projectile fired at an angle equally inclined to the horizontal and vertical with velocity u, findthe horizontal range?

8. A body is so projected in the air that the horizontal range covered by the body is equal to the vertical heightattained by the body. Find the angle of projection ?

9. What should be the angles of projection to obtain maximum height and maximum time of flight ?

10. A cricketer can throw a ball to a maximum horizontal distance of 100 m How much high should above theground can the cricketer thrown the same ball.

11. What are horizontal & vertical components of acceleration of a body thrown horizontally with uniform speed.

12. A bob of mass 0.1 kg hung from the ceiling of a room by a string 2 m long is set into oscillation. The speedof the bob at its mean position is 1 ms–1. What is the trajectory of the bob ; if the string is cut(a) when the bob is at one of its extreme position(b) at its mean position

13. A projectile can have the same range R for two angles of projections. If t1 & t2 be the times of flight in twocases, then find out relation between t1t2 and R ?

14. The height y and the distance x along the horizontal plane of a projectile on a certain planet (without nosurrounding atmosphere) are given by y = (8t – 5t2) metere & x = 6t m. Then what will be the velocity ofprojection ?

15. Find out the time after which the velocity of the particle becomes perpendicular to the initial velocity.

Section : B Projectile from a tower

16. A glass marble projected horizontal from the top of a table falls at a distance x from the edge of the table. Ifh is the height of the table, find the velocity of projection ?

17. A ball is thrown horizontally and at the same time another ball is dropped down from the top of a tower (a) willboth the balls reach the ground at the same time. (b)Will both strike at the ground with same velocity ?

18. Observe the figure carefully & answer the following :

(a) All the particles thrown with same initial velocity would strike the ground with(same/different) speed at (different/same) time irrespective of their initial direction of velocity

(b) Time would be least for particle (1 or 2)

19. A projectile is fired horizontally with a velocity of 98 m/s from the top of a hill 490 m high. Find :(i) the time taken to reach the ground(ii) the distance of the target from the hill(iii) the velocity with which the particle hits the ground

Section : C Equation of trajectory

20. The equation of a projectile is y = 3 x –2

gx2

find the angle of projection ?

Section : D Projectile from a moving object

21. Consider a boy on a trolley who throws a ball with speed 20 m/s at an angle 370 w.r.t. trolley which moveshorizontally with speed 10 m/s.

(a) Find horizontal and vertical components of initial velocity of ball when ball is projected in direction ofmotion of trolley.

(b) Find horizontal and vertical components of initial velocity of ball when ball is projected opposite todirection of motion of trolley

22. Consider a boy on a platform who throws a ball with speed 20 m/s at an angle 370 w.r.t. platform which movesupwards with speed 10 m/s. (a) Find the horizontal and vertical component of balls velocity.

(b) Find horizontal and vertical components of balls velocity when ball is projected downwards from the platform.

23. A bomb is dropped from an aeroplane flying horizontally with a velocity 469 m s–1 at an altitude of 980 m. Findthe time after which bomb will hit the ground?

24. A bomb is dropped from an aeroplane moving horizontally at a certain height from the ground. Does the timetaken by the bomb to reach the ground depend on the velocity of the aeroplane ?

25. A bomb is released from a horizontal flying bomber when it is vertically above the target, shall it hit the target?

Section : E Projectile on an inclined plane

26. A particle is projected at an angle with an inclined plane making an angle with the horizontal as shownin figure, speed of the particle is u, after time t find :

(a) x component of acceleration ?(b) y component of acceleration ?(c) x component of velocity ?(d) y component of velocity ?(e) x component of displacement ?(f) y component of displacement ?(g) y component of velocity when particle is at maximum distance from the incline plane ?

27. On an inclined plane of inclination 30º, a ball is thrown at an angle of 60º with the horizontal from the foot of

the incline with a velocity of 10 3 ms–1. If g = 10 ms–2, then find the time in which ball will hit the inclinedplane?

28. A particle is moving eastward with a velocity of 5ms-1. In 10second the velocity changes to 5ms–1 northwards.Find the average acceleration in this time ?

29. A body starts from rest from the origin with an acceleration of 6 ms–2 along the x-axis and 8 ms–2 along the y-axis. Find the distance from the origin after 4 seconds ?

Section : A Definition, Projectile on a horizontal plane1. A point mass is projected, making an acute angle with the horizontal. If angle between velocity v

and

acceleration g

is , then is given by(A) 90º < 0 < 0º (B) = 90º (C) = 90º (D) 0º < < 180º

2. Two stones are projected from the same point with same speed making angles 45º + and 45º–with thehorizontal respectively. If < 45º, then the horizontal ranges of the two stones are in the ratio of(A) 1 : 1 (B) 1 : 2 (C) 1 : 3 (D) 1 : 4

3. A hunter takes an aim at a monkey sitting an a tree and fires a bullet. Just when the bullet leaves barrel of thegun, it so happens that the monkey begins to fall freely. The bullet will(A) go above the monkey(B) go below the monkey(C) hit the monkey(D) may or may not hit the monkey. It will depend upon the velocity of the bullet.

4. It was calculated that a shell when fired form a gun with a certain velocity and at an angle of elevation 365

rad

should strike a given target. In actual practice, it was found that a hill just prevented the trajectory. At whatangle of elevation should the gun be fired to hit the target.

(A) 365

rad (B) 3611

rad (C) 367

rad (D) 3613

rad.

5. A projectile is thrown with a speed v at an angle with the vertical. Its average velocity between the instantsit crosses half the maximum height is(A) v sin , horizontal and in the plane of projection(B) v cos , horizontal and in the plane of projection(C) 2v sin , horizontal and perpendicular to the plane of projection(D) 2v cos , vertical and in the plane of projection.

6. A bullet is fired horizontally from a rifle at a distant target. Ignoring the effect of air resistance, which of thefollowing is correct?

Horizontal VerticalAcceleration Acceleration

(A) 10 ms–2 10 ms–2

(B) 10 ms–2 0 ms–2

(C) 0 ms–2 10 ms–2

(D) 0 ms–2 0 ms–2.

7. If the initial velocity of a projectile be doubled, keeping the angle of projection same, the maximum heightreached by it will(A) be halved (B) be quadrupled (C) be doubled (D) remain the same.

8. Two bullets are fired horizontally, simultaneously and with different velocities from the same place. Whichbullet will hit the ground earlier?(A) It would depend upon the weights of the bullets (B) The slower one.(C) The faster one (D) Both will reach simultaneously.

10. A stone is thrown upwards. It returns to ground describing a parabolic path. Which of the following remainsconstant?(A) speed of the ball (B) kinetic energy of the ball(C) vertical component of velocity (D) horizontal component of velocity.

Section : B Projectile from a tower

11. A body is thrown horizontally with a velocity gh2 from the top of a tower of height h. It strikes the levelground through the foot of the tower at a distance x from the tower. The value of x is

(A) h (B)2h

(C) 2h (D) 3h2

12. One stone is projected horizontally from a 20 m high cliff with an initial speed of 10 ms–1. A second stone issimultaneously dropped from that cliff. Which of the following is true?(A) Both strike the ground with the same velocity.(B) The ball with initial speed 10 ms–1 reaches the ground first.(C) Both the balls hit the ground at the same time.(D) One cannot say without knowing the height of the building.

Section : C Equation of trajectory13. A particle, with an initial velocity v0 in a plane, is subjected to a constant acceleration in the same plane.

Then, in general, the path of the particle could be(A) a circle (B) a straight line (C) a parabola (D) a hyperbola.

14. A ball is projected from a certain point on the surface of a planet at a certain angle with the horizontal surface.The horizontal and vertical displacement x and y vary with time t in second as:

x = 10 t3 and y = 10t – t2

The maximum height attained by the ball is(A) 100 m (B) 75 m (C) 50 m (D) 25 m.

Section : D Projectile from a moving object15. A bag is dropped from an aeroplane flying horizontally at a constant speed. If air resistance is ignored, where

will the aeroplane be when the bag hits the ground?(A) ahead of the bag (B) directly above the bag(C) far behind the bag (D) data is not sufficient.

16. An aeroplane flying horizontally with a speed of 360 km h–1 releases a bomb at a height of 490 m from theground. When will the bomb strike the ground?(A) 8 s (B) 6 s (C) 7 s (D) 10 s.

17. The path of one projectile in motion as seen from another moving projectile is(A) a straight line (B) a circle (C) an ellipse (D) a parabola.

Section : E Projectile on an inclined plane18. A plane surface is inclined making an angle with the horizontal. From the bottom of this inclined plane, a

bullet is fired with velocity v. The maximum possible range of the bullet on the inclined plane is

(A) gv2

(B) )sin1(gv2

(C) )sin1(gv2

(D) )cos1(gv2

19. A ball is projected horizontal with a speed v from the top of a plane inclined at an angle 45º with thehorizontal. How far from the point of projection with the ball strike the plane?

(A) gv2

(B) 2 gv2

(C) gv2 2

(D) 2

gv2 2

20. The time of flight of a projectile on an upward inclined plane depends upon(A) angle of inclination of the plane (B) angle of projection(C) the value of acceleration due to gravity (D) all of these.

1. A particle moves in the xy plane with only an x-component of acceleration of 2 m s–2. The particle starts fromthe origin at t = 0 with an initial velocity having an x-component of 8 m s–1 and y-component of –15 ms–1.Velocity of particle after time t is :

(A) [(8 + 2t) i – 15 j ] m s–1 (B) zero

(C) 2t i + 15 j (D) directed along z-axis.

2. A ball rolls of the top of a stairway horizontally with a velocity of 4.5 m s–1. Each step is 0.2 m high and 0.3m wide. If g is 10 ms–1, then the ball will strike the nth step where n is equal to(A) 9 (B) 10 (C) 11 (D) 12.

3. A plane flying horizontally at a height of 1500 m with a velocity of 200 ms–1 passes directly overhead onantiaircraft gun. Then the angle with the horizontal at which the gun should be fired for the shell with a muzzlevelocity of 400 m s–1 to hit the plane, is(A) 90º (B) 60º (C) 30º (D) 45º

4. The velocity of projection of a projectile is (6 i + 8 j ) ms–1.. The horizontal range of the projectile is(A) 4.9 m (B) 9.6 m (C) 19.6 m (D) 14 m.

5. A ball is projected from the ground with velocity 25 ms–1. Two seconds later, ball crosses a wall 5m high.What is the angle of projection of the stone? (Take g = 10 ms–2)(A) 75º (B) 60º (C) 45º (D) 30º.

6. A body has an initial velocity of 3 ms–1 and has an acceleration of 1 ms–2 normal to the direction of the initialvelocity. Then its velocity, 4 second after the start is(A) 7 ms–1 along the direction of initial velocity(B) 7 ms–1 along the normal to the direction of the initial velocity(C) 7 ms–1 mid-way between the two directions

(D) 5 ms–1 at an angle of tan-143

with the direction of the initial velocity..

7. If R and h represent the horizontal range and maximum height respectively of an obliquie projectile, then

h8R2

+ 2h represents

(A) maximum horizontal range (B) maximum vertical range(C) time of flight (D) velocity of projectile at highest point

8. A projectile is thrown with velocity v making an angle with the horizontal. It just crosses the top of two poles,each of height h, after 1 second and 3 second respectively. The time of flight of the projectile is(A) 1 s (B) 3 s (C) 4 s (D) 7.8 s.

9. In the above q. maximum height attained by the projectile is(A) g (B) 2g (C) 3g (D) 4g

10. In above Q. h is

(A)2g

(B) g (C)2g3

(D) 2g.

11. Two tall buildings are 40 m apart. With what speed must a ball be thrown horizontally from a window 145 mabove the ground in one building, so that it will enter a window 22.5 m from the ground in the other?(A) 5 ms–1 (B) 8 ms–1 (C) 10 ms–1 (D) 16 ms–1.

12. A particle move along the parabolic path x = y2 + 2y + 2 in such a way that the y-component of velocityvector remain 5m/s during the motion. The magnitude of the accleration of the particle is :

(A) 50 m/s2 (B) 100 m/s2 (C) 10 2 m/s2 (D) 0.1 m/s2

13. A ball is projected from point A with a velocity 10 m/s perpendicular to the inclined plane as shown infigure. Range of the ball on the inclined plane is :

(A) 340

m (B) 1320

m (C) 2013

m (D) 4013

m

14. A particle is projected from a horizontal plane with 8 2 m/s at an angle. At highest point its velocityis found to be 8 m/s. Its range will be :(A) 64 m (B) 12.8 m (C) 128 m (D) 64.4 m

15. A projectile is projected at an angle (> 45°) with an initial velocity u. The time [t] at which its horizontalcomponent will equal the vertical component :

(A) t = gu

( cos – sin ) (B) t = gu

( cos + sin )

(C) t = gu

( sin – cos ) (D) t = gu

( sin2 – cos2 )

1. A player kicks a football at an angle of 45º with an initial speed of 20 m/s . A second player on the goalline 60 m away in the direction of kick starts running to receive the ball at that instant. Find the speedof the second player with which he should run to catch the ball before it hits the ground.[g = 10 m/s2 ]

2. If at a point of the parabolic path the velocity be u and the inclination to the horizon be , at what timethe particle is moving at right angles to its former direction.

3. The direction of motion of a projectile at a certain instant is inclined at an angle to the horizon. Aftert seconds it is inclined an angle . Find the horizontal component of velocity of projection in terms ofg, t , and

4. A radius vector of a point A relative to the origin varies with time t as r = j2tbita , where a and b

are positive constants and jandi are the unit vectors of the x and y axes. Find:(i) the equation of the point's trajectory y (x) ; plot this function(ii) the time dependence of the velocity v and acceleration a vectors as well as of the moduli of

these quantities .

5. Two particles are projected simultaneously with the same speed V in the same vertical plane withangles of elevation and 2 , where < 45º . At what time will their velocities be parallel.

6. If 4 seconds be the time in which a projectile reaches a point P of its path and 5 seconds the time fromP till it reaches the horizontal plane through the point of projection . Find the height of P above thehorizontal plane. [ g = 9.8 m/sec2 ]

7. From the top of a tower of height 40 m, a ball is projected upwards with a speed of 20 m/sec at anangle of 30º to the horizontal. When and at what distance from the foot of the tower does the ball hit theground ? What is the speed of the ball at this instant ?

8. A Bomber flying upward at an angle of 53º with the vertical releases a bomb at an altitude of 800 m. Thebomb strikes the ground 20 sec after its release . Find :(i) the velocity of the bomber at the time of release of the bomb .(ii) the maximum height attained by the bomb .(iii) the horizontal distance travelled by the bomb before it strikes the ground(iv) the velocity (magnitude and direction) of the bomb just when it strikes the ground .

[ G i v e n s i n 5 3 º = 0 . 8 ; g = 1 0 m s

2]

9. A man is travelling on a flat car which is moving up a plane inclined atcos = 4/5 to the horizontal with a speed 5 m/s. He throws a ball towardsa stationary hoop located perpendicular to the incline in such a way thatthe ball moves parallel to the slope of the incline while going through thecentre of the hoop. The centre of the hoop is 4 m high from the man's handcalculate the time taken by the ball to reach the hoop.

Questions for short answers.10. You are driving directly behind a pickup truck, going at the same speed as the truck. A crate falls from

the bed of the truck to the road. Will your car hit the crate before the crate hits the road if you neitherbrake nor swerve?

11. What doesd vd t

and

dvd t

represent? Can these be equal? Can:

(a)d vd t

= 0 while

dvd t

0 (b)

d vd t

0 while

dvd t

= 0

12. Is it possible for a body to change the direction of its velocity vector when the body is experiencingconstant acceleration?

13. Is it important in the long jump that how much height you take for jumping?

14. No matter what the initial velocity, the motion of a projectile (neglecting air resistance) is alwaysconfined to a single plane. Why?

1. A ship is approaching a cliff of height 105 m above sea level. A gun fitted on the ship can fire shots witha speed of 110 ms1. Find the maximum distance from the foot of the cliff from where the gun can hit anobject on the top of the cliff. [ g = 10 m/s2 ] [REE '94, 6]

2. Two towers AB and CD are situated a distance ' d ' apart as shown in the fig.AB is 20 m high and CD is 30 m high from the ground. An object of mass ' m'is thrown from the top of AB horizontally with a velocity of 10 m/s towards CD.Simultaneously another object of mass 2 m is thrown from the top of CD at anangle of 60º to the horizontal towards AB with the same magnitude of initialvelocity as that of the first object. The two objects move in the same vertical plane, collide in mid airand stick to each other.(i) calculate the distance ' d ' between the towers.(ii) find the position where the objects hit the ground. [JEE ' 94, 6]

3. A building 4.8 m high 2b meters wide has a flat roof. A ball is projected from a point on the horizontalground 14.4 m away from the building along its width. If projected with velocity 16 m/s at an angleof 45º with the ground, the ball hits the roof in the middle, find the width 2b. Also find the angle of

projection so that the ball just crosses the roof if projected with velocity 10 3 m/s. [REE '95, 6]

4. Two guns situated on the top of a hill of height 10 m, each fired shots with the same speed 5 3 m/sat some interval of time. One gun fires horizontally and other fires upwards at an angle of 60º with thehorizontal. The shot collide in air at a point P. Find:(i) the time interval between the firings and(ii) the coordinates of the point P.Take origin of the coordinate system at the foot of the hill right below the muzzle and trajectories inx-y plane. [JEE '96, 5]

5. A vertical pole has a red mark at some height. A stone is projected from a fixed point on the ground.When projected at an angle of 45º it hits the pole orthogonally 1 m above the mark. When projectedwith a different velocity at an angle of tan1(3/4), it hits the pole orthogonally 1.5 m below the mark. Findthe velocity and angle of projection so that it hits the mark orthogonally to the pole. [ g = 10 m/sec2 ]

[REE '96, 6 ]

6. The trajectory of a projectile in a vertical plane is y = ax bx2, where a, b are constants and x and y arerespectively the horizontal and vertical distances of the projectile from the point of projection. Themaximum height attained is _______ and the angle of projection from the horizontal is _______.

[JEE '97, 2]

7. The coordinates of a particle moving in a plane are given by x (t) = a cos (p t) andy (t) = b sin (p t), where a, b (< a) and p are positive constants of appropriate dimensions.(A) the path of the particle is an ellipse(B) the velocity and acceleration of the particle are normal to each other at t = /2p(C) the acceleration of the particle is always directed towards a focus(D) the distance travelled by the particle in time interval t = 0 to = /2p is a. [JEE '99, 2]

8. An object A is kept fixed at the point x = 3 m and y = 1.25 m on a plankP raised above the ground. At time t = 0 the plank starts moving along the+ x direction with an acceleration 1.5 m/s2. At the same instant a stone isprojected from the origin with a velocity u as shown. A stationary personon the ground observes the stone hitting the object during its downwardmotion at an angle of 45º to the horizontal. All the motions are in x-yplane. Find ‘u’ and the time after which the stone hits the object. [ Take g= 10 m/s2 ]

[JEE 2000, 5]9. Shots fired simultaneously from the top and foot of a vertical cliff at elevations of 300 and 600 respectively,

strike an object simultaneously which is at a height of 100 meters from the ground and at a horizontal

distance of 200 3 meters from the cliff. Find the height of the cliff, the velocities of projection of theshots and the time taken by the shots to hit the object. [ REE ’2000, 5 ]

Exercise - 1

1. It is projected at an angle other than 90º with thehorizontal

2. Angle of projection = 450 . 3. 900 .

4. At the highest point.

5. sin : cos 6. sin2 : cos2

7. gu2

8. tan = 4.

9. (a) = 900 , (b) = 900 .10. 50 m

11. (a) 0, (b) g

12. (a) vertically downwards (b) parabolic path

13. t1t2 = 2R/g

14. 10 m/s 15. 34

16. xh2g

17. (a) yes (b) no

18. (a) same, different (b) 1

19. (i) 10 sec. (ii) 980 m (iii) 138.59 m/s

20. tan = 321. (a) 26, 12 (b) –6, 12

22. (a) 16, 22 (b) 16, 2

23. 10 2 s 24. No

25. No

26. (a) – g sin, (b) –g cos, (c) u cos – g sin × t,(d) u sin – gcos × t,

(e) u cos× t –21

g sin× t2, (f) u sin× t –21

g

cos× t2, (g) zero.

27. 2 s

28. 21

ms–2 towards north-west 29. 80 m

EXERCISE-21. D 2. A 3. C 4. D 5. A 6. C

7. D 8. D 10. D 11. C 12. C 13. C

14. D 15. B 16. D 17. A 18. B 19. D

20. D

Exercise-3

1. A 2. A 3. B 4. B 5. D 6. D

7. A 8. C 9. B 10. C 11. B 12. A

13. A 14. B 15. C

Exercise-4

1. 5 2 m/s 2. (u/g) cosec .

3. tantangt

4. (i) y = bxa

2

2

(ii)v = a i b t j 2 ,

= 2 b j ,

v = a b t2 2 24 ,

= 2 b

5.

23eccos

2cos

gv

6. 98 meters

7. 4 s, 40 3 m, 20 3 m/s

8. (i) 100 m/s (ii) 980 m (iii) 1600 m(iv) = tan-1 (7/4) = 60º 15',

v = 20 65 =161.245 m/s

9. 1 s

Exercise-5

1. 1100 m2. (i) 17.32 m

(ii) combined mass strikes at 11.55 m from B and 5.77 m from D

3. width of the roof is 9.6 m; = tan1

23

4. (i) 1 s (ii) ( 5,35 )

5.3

3620m/s , tan1

109

6. b4a2

, tan1 a 7. A, B

8. u = 7.29 m/s, t = 1 s.

9. 400 m, VT = 40 3 m/s, VF = 40 m/s, T = 10 s.