Motion in a plane

1. A football is kicked with a velocity of 20 𝑚/𝑠 at an angle of 45° with the horizontal. (a) Find
the time taken by the ball to strike the ground. (b) Find the maximum height it reaches. (c) How
far away from the kick does it hit the ground? Take 𝑔 = 10 𝑚/𝑠 2 .𝟐√𝟐 𝒔, 𝟏𝟎 𝒎, 𝟒𝟎 𝒎

2. A helicopter on flood relief mission, flying horizontally with a speed u at an altitude H, has to
drop a food packet for a victim standing on the ground. At what distance from the victim should
the packet be dropped? The victim stands in the vertical plane of the helicopter's motion.
𝟐𝒖𝟐 𝑯
√ + 𝑯𝟐
𝒈
3. A particle is projected horizontally with a speed 𝑢 from the top of a plane inclined at an angle 𝜃
with the horizontal. How far from the point of projection will the particle strike the plane?
𝟐𝒖𝟐
𝐭𝐚𝐧 𝜽 𝐬𝐞𝐜 𝜽
𝒈
4. At 𝑡 = 0, truck, starting from rest, moves in the positive x-direction at uniform acceleration of
5 𝑚𝑠 −2 . At 𝑡 = 20 𝑠, a ball is released from the top of the truck. The ball strikes the ground in
1 𝑠 after the release. Find the veloity of the ball, when it strikes the ground. 𝟏𝟎𝟎 𝒊̂ − 𝟏𝟎 𝒋̂

5. A projectile is fired with a speed u at an angle 𝜃 with the horizontal. Find its speed when its
direction of motion makes an angle 𝛼 with the horizontal. 𝒖 𝐜𝐨𝐬 𝜽 𝐬𝐞𝐜 𝜶

6. A bullet is fired horizontally aiming at an object which starts falling at the instant the bullet is
fired. Show that the bullet will hit the object.
7. A ball is thrown at a speed of 40 𝑚/𝑠 at an angle of 60° with the horizontal. Find (a) the
maximum height reached and (b) the range of the ball. Take 𝑔 = 10 𝑚/𝑠 2
.(a)𝟔𝟎 𝒎, (b)𝟖𝟎 √𝟑 𝒎
8. A ball is thrown horizontally from a point 100 𝑚 above the ground with a speed of 20 𝑚/𝑠. Find
(a) the time it takes to reach the ground, (b) the horizontal distance it travels before reaching the
ground, (c) the velocity (direction and magnitude) with which it strikes the ground. (a) 𝟒. 𝟓 𝒔 (b)
𝟗𝟎 𝒎 (c) 𝟒𝟗 𝒎/𝒔, 𝜽 = 𝟔𝟔° with horizontal.

9. Find the average velocity of a projectile between the instants it crosses half the maximum height.
It is projected with a speed u at an angle 𝜃 with the horizontal.
10. A police inspector in a jeep is chasing a pickpocket on a straight road. The jeep is going at its
maximum speed 𝑣 (assumed uniform). The pickpocket rides on the motorcycle of a waiting friend
when the jeep is at a distance 𝑑 away, and the motorcycle starts with a constant acceleration a.
Show that the pickpocket will be caught if 𝑣 ≥ √2𝑎𝑑

11. A particle moves in the 𝑋 − 𝑌 plane with a constant acceleration of 1.5 𝑚/𝑠 2 in the direction
making an angle of 37° with the X-axis. At 𝑡 = 0 the particle is at the origin and its velocity is
8.0 𝑚/𝑠 along the X-axis. Find the velocity and the position of the particle at 𝑡 = 4.0 𝑠.
𝟏𝟑. 𝟑 𝒎/𝒔 , (𝟒𝟏. 𝟔 𝒎, 𝟕. 𝟐 𝒎)
12. Position of an ant (S in metres) moving in Y-Z plane is given by 𝑆⃗ = 2𝑡 2 𝑗̂ + 5 𝑘̂ (where 𝑡 is in
second). What will be the magnitude and direction of velocity of the ant at 𝑡 = 1 𝑠 . 𝟒 𝒎/𝒔 in
+𝒚 direction.
13. The trajectory of projectile, projected from the ground is given by 𝑦 = 𝑥 − 20.
𝑥2

Where 𝑥 and 𝑦 are measured in meter. What will be the maximum height attained by the
projectile. 𝟓 𝒎
14. Two objects are projected with same velocity ′𝑢′ however at different angles 𝛼 and 𝛽 with the
horizontal. If 𝛼 + 𝛽 = 90°,what will be the ratio of horizontal range of the first object to the 2nd
object? 𝟏: 𝟏
15. At time 𝑡 = 0 a particle starts travelling from a height 7 𝑧̂cm in a plane keeping z coordinate
constant. At any instant of time, it's position along the 𝑥̂ and 𝑦̂ directions are defined
as 3𝑡 and 5𝑡 3 respectively. At 𝑡 = 1𝑠, what will be the acceleration of the particle? 𝟑𝟎 𝒚 ̂

16. A projectile is launched at an angle ′𝛼′ with the horizontal with a velocity 20 𝑚𝑠 −1. After 10 𝑠,
its inclination with horizontal is '𝛽'. Find the value of tan 𝛽. (𝑔 = 10 𝑚𝑠 −2 ). 𝐭𝐚𝐧 𝜶 − 𝟓 𝐬𝐞𝐜 𝜶

17. A butterfly is flying with a velocity 42 𝑚/𝑠 in North-East direction. Wind is slowly blowing at
1 𝑚/𝑠 from North to South. Find the resultant displacement of the butterfly in 3 seconds.𝟏𝟓 𝒎

18. A mosquito is moving with a velocity 𝑣⃗ = 0.5𝑡 2 𝑖̂ + 3𝑡 𝑗̂ + 9 𝑘̂ 𝑚/𝑠 and accelerating in uniform
√𝟖𝟓
conditions. What will be the direction of mosquito after 2 𝑠? 𝐭𝐚𝐧−𝟏 ( ) from y-axis
𝟔
19. The trajectory of a projectile in a vertical plane is 𝑦 = 𝛼𝑥 − 𝛽𝑥 2 , where 𝛼 and 𝛽 are constants
and 𝑥 & 𝑦 are respectively the horizontal and vertical distances of the projectile from the point
of projection. Find the angle of projection 𝜃 and the maximum height attained H respectively?
𝜶𝟐
𝐭𝐚𝐧−𝟏 𝜶 , 𝟒𝜷
20. Starting from the origin at time 𝑡 = 0, with initial velocity 5 𝑗̂ 𝑚𝑠 −1 , a particle moves in the x-
y plane with a constant acceleration of (10 𝑖̂ + 4 𝑗̂) 𝑚𝑠 −2 . At time t, its coordinates are
(20 𝑚, 𝑦0 𝑚). What are the values of 𝑡 and 𝑦0 respectively. 𝟐 𝒔, 𝟏𝟖 𝒎
21. A particle moves such that its position vector 𝑟⃗ (𝑡) = 𝑐𝑜𝑠 𝜔𝑡 𝑖̂ + 𝑠𝑖 𝑛 𝜔𝑡 𝑗̂ where 𝜔 is a constant
and t is time. Then which of the following statements is true for the velocity 𝑣⃗(𝑡) and
acceleration 𝑎⃗(𝑡) of the particle:
(A) 𝑣⃗ and 𝑎⃗ both are perpendicular to 𝑟⃗
(B) 𝑣⃗ and 𝑎⃗ both are parallel to 𝑟⃗
(C) ⃗𝒗⃗ is perpendicular to ⃗𝒓⃗ and ⃗𝒂⃗ is directed towards the origin
(D) 𝑣⃗ is perpendicular to 𝑟⃗ and 𝑎⃗ is directed away from the origin