1 Mark Questions

1. What is the angle between the direction of velocity and acceleration at the highest point of a projectile
path?
2. Is the maximum height attained by projectile largest when its horizontal range is maximum?
3. What is the angle of projection for a projectile motion whose range R is 𝑛 times the maximum height H?

3 Marks Questions

1. A projectile is fired horizontally with a velocity 𝑢. Show that its trajectory is a parabola. Also obtain
expressions for its (i) time of flight, (ii) horizontal range and (iii) velocity at any instant.
2. At which points on the projectile trajectory is the (i) potential energy maximum, (ii) kinetic energy
maximum and (iii) total energy maximum?

5 Marks Questions

1. A projectile is fired with a velocity 𝑢 making an angle 𝜃 with the horizontal. Show that its trajectory is a
parabola. Derive expressions for (i) time of maximum height (ii) time of flight, (iii) maximum height, (iv)
horizontal range. Show that there are two angles of projections for the same horizontal range. Also
calculate the velocity of the projectile at any instant 𝑡 and at the end point of the flight.

Numericals

1. A hiker stands on the edge of a cliff 490m above the ground and throws a stone horizontally with an
initial speed of 15ms-1. Neglecting air resistance, find the time take by the stone to reach the ground,
and the speed with which it hits the ground. (Take g = 9.8ms -2)
2. A projectile is fired horizontally with a velocity of 98 ms-1 from the top of a hill 490m high. Find (i) the time
taken to reach the ground (ii) the distance of the target from the hill and (iii) the velocity with which the
projectile hits the ground.
3. A body is thrown horizontally from the top of a tower and strikes the ground after three seconds at an
angle of 45o with the horizontal. Find the height of the tower and the speed with which the body was
projected. Take g = 9.8ms-2
4. A body is projected horizontally from the top of a cliff with a velocity of 9.8ms -1. What time elapses
before horizontal and vertical velocities become equal? Take g = 9.8ms -2
5. A particle is projected horizontally with a speed 𝑢 from the top of plane inclined at an angle 𝜃 with the
horizontal. How far from the point of projection will the particle strike the plane?
6. A helicopter of a flood relief mission flying horizontally with a speed 𝑢 at an altitude ℎ, has to drop a food
packet for victim standing on the ground. At what distance from the victim should the food packet be
dropped?
7. Two tall buildings are situated 200m apart. With what speed must a ball be thrown horizontally from the
window 540m above the ground in one building, so that it will enter a window 50m above the ground in
the other?
8. In between two hills of heights 100m and 92m respectively, there is a valley of breadth 16m. If a vehicle
jumps from the first hill to the second, what must be its minimum horizontal velocity so that it may not
fall into the valley? Take g = 9 ms-2
9. A projectile has a range of 50m and reaches a maximum height of 10m. Calculate the angle at which the
projectile is fired.
10. A boy stands at 39.2m from a building and throws a ball which just passes through a window 19.6m
above the ground. Calculate the velocity of projection of the ball.
11. Find the angle of projection for which the horizontal range and the maximum height are equal.
12. Prove that the maximum horizontal range is four times the maximum height attained by the projectile
when fired at an inclination so as to have maximum horizontal range.
13. Show that a given gun will shoot three times as high when elevated at an angle of 60o as when fired at
angle of 30o but will carry the same distance on a horizontal plane.
14. A ball is thrown at an angle 𝜃 and another ball is thrown at angle (90° − 𝜃) with the horizontal direction
from the same point with a velocity of 39.2 ms-1. The second ball reaches 50m higher that the first ball.
Find their individual heights. Take g = 9.8ms -2
15. If R is the horizontal range for 𝜃 inclination and ℎ is the maximum height reached by the projectile, show
𝑅2
that the maximum range is given by + 2ℎ
8ℎ
16. Show that there are two angles of projection for which the horizontal range is the same. Also show that
the sum of the maximum heights for these two angles is independent of the angle of projection.
17. Show that there are two values of time for which a projectile is at the same height. Also show that the
sum of these two times is equal to the time of flight.
18. Two projectiles are thrown with different velocities and at different angles so as to cover the same
maximum height. Show that the sum of the times taken by each to reach the highest point is equal to
the total time taken by either of the projectiles.
19. At what angle should a body be projected with a velocity 24ms-1 just to pass over the obstacle 16m high
at a horizontal of 32m? Take g = 10ms-2
20. A particle is projected over a triangle from one end of a horizontal base and grazing the vertex falls on
the other end of the base. If 𝛼 and 𝛽 be the base angles and 𝜃 the angle of projection, prove that tan 𝜃 =
tan 𝛼 + tan 𝛽
21. A football is kicked at 20ms-1 at a projection angle of 45o. A receiver on the goal line 25 metres away in
the direction of the kick runs at the same instant to meet the ball. What must be his speed, if he is to
catch the ball before it hits the ground?
22. Prove that the time of flight T and the horizontal range R of a projectile are connected by the equation,
𝑔𝑇 2 = 2𝑅 tan 𝜃 where 𝜃 is the angle of projection.
23. A body is projected such that its kinetic energy at the top is 3/4th of its initial kinetic energy. What is the
initial angle of projection of the projectile with the horizontal?