Impulse & momentum

The points A, B and C lie on a smooth horizontal plane. A small ball of mass 0.2 kg is moving along the line AB with speed 4 m s^–1. When the ball is at B, the ball is given an impulse. Immediately after the impulse is given, the ball moves along the line BC with speed 7 m s^–1. The line BC makes an angle of 35° with the line AB, as shown in Figure 1.

Find the magnitude of the impulse given to the ball.

Find the size of the angle between the direction of the impulse and the original direction of motion of the ball.

A particle P of mass 0.5 kg is moving with velocity 4j m s^–1 when it receives an impulse I Ns. Immediately after P receives the impulse, the velocity of P is (2i + 3j) m s^–1. Find

the magnitude of I,

the angle between I and j.

Two particles A and B, of masses 3 m and 4 m respectively, lie at rest on a smooth horizontal surface. Particle B lies between A and a smooth vertical wall which is perpendicular to the line joining A and B. Particle B is projected with speed 5 u in a direction perpendicular to the wall and collides with the wall. The coefficient of restitution between B and the wall

Find the magnitude of the impulse received by B in the collision with the wall.

After the collision with the wall, B rebounds from the wall and collides directly with A. The coefficient of restitution between A and B is e.

Show that, immediately after they collide, A and B are both moving in the same direction.

The kinetic energy of B immediately after it collides with A is one quarter of the kinetic energy of B immediately before it collides with A.

Find the value of e.

A particle of mass 0.6kg is moving with constant velocity (ci + 2cj)ms, where c

Immediately after receiving the impulse the particle has velocity (2ci – cj)ms. Find the value of c.

Two particles A and B, of mass 2 m and 3 m respectively, are initially at rest on a smooth horizontal surface. Particle A is projected with speed 3u towards B. Particle A collides directly with particle B. The coefficient of restitution between A and B is

Find

the speed of A immediately after the collision,

the speed of B immediately after the collision.

After the collision B hits a fixed smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of B. The coefficient of restitution between B and the wall is e. The magnitude of the impulse received by B when it hits the wall is

Find the value of e.

Determine whether there is a further collision between A and B after B rebounds from the wall.

A ball of mass 0.4 kg is moving in a horizontal plane when it is struck by a bat. The bat exerts an impulse (–5 i + 3 j) N s on the ball. Immediately after receiving the impulse the ball has velocity (12i + 15j) m s^–1. Find

the speed of the ball immediately before the impact,

the size of the angle through which the direction of motion of the ball is deflected by the impact.

A particle P of mass 2 m is moving in a straight line with speed 3 u on a smooth horizontal table. A second particle Q of mass 3 m is moving in the opposite direction to P along the same straight line with speed u. The particle P collides directly with Q. The direction of motion of P is reversed by the collision. The coefficient of restitution between P and Q is e.

Show that the speed of Q immediately after the collision is 8e + 3)

Find the range of possible values of e.

The total kinetic energy of the particles before the collision is T. The total kinetic energy of the particles after the collision is kT. Given that e =

find the value of k.

A particle P of mass 2 kg is moving with velocity (i – 4j) m s^–1 when it receives an impulse of (3 i + 6 j) N s. Find the speed of P immediately after the impulse is applied.

Two particles P and Q, of masses 2 m and m respectively, are on a smooth horizontal table. Particle Q is at rest and particle P collides directly with it when moving with speed u. After the collision the total kinetic energy of the two particles is mu². Find

the speed of Q immediately after the collision,

the coefficient of restitution between the particles.

Three particles P, Q and R lie at rest in a straight line on a smooth horizontal table with Q between P and R. The particles P, Q and R have masses 2 m, 3 m and 4 m respectively. Particle P is projected towards Q with speed u and collides directly with it. The coefficient of restitution between each pair of particles is e.

Show that the speed of Q immediately after the collision with P is (1 + e)u.

After the collision between P and Q there is a direct collision between Q and R.

the speed of Q after this collision,

the speed of R after this collision.

Immediately after the collision between Q and R, the rate of increase of the distance between P and R is V.

Find V in terms of u.

A small ball B of mass 0.25 kg is moving in a straight line with speed 30 m s^–1 on a smooth horizontal plane when it is given an impulse. The impulse has magnitude 12.5 N s and is applied in a horizontal direction making an angle of (90° + ), where tan , with the initial direction of motion of the ball, as shown in Figure 3.

Find the speed of B immediately after the impulse is applied.

Find the direction of motion of B immediately after the impulse is applied.

A ball of mass 0.5 kg is moving with velocity 12i m s^–1 when it is struck by a bat. The impulse received by the ball is (–4 i + 7 j) N s. By modelling the ball as a particle, find

the speed of the ball immediately after the impact,

the angle, in degrees, between the velocity of the ball immediately after the impact and the vector i,

the kinetic energy gained by the ball as a result of the impact.

A particle of mass 2 kg is moving with velocity (5i + j) m s^–1 when it receives an impulse of (–6 i + 8 j) N s. Find the kinetic energy of the particle immediately after receiving the impulse.

[ In this question i and j are perpendicular unit vectors in a horizontal plane. ] A ball of mass 0.5 kg is moving with velocity (10i + 24j) m s^–1 when it is struck by a bat. Immediately after the impact the ball is moving with velocity 20i m s^–1. Find

the magnitude of the impulse of the bat on the ball,

the size of the angle between the vector i and the impulse exerted by the bat on the ball,

the kinetic energy lost by the ball in the impact.

Two particles, P, of mass 2 m, and Q, of mass m, are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide. Immediately before the collision the speed of P is 2 u and the speed of Q is u. The coefficient of restitution between the particles is e, where e < 1. Find, in terms of u and e,

the speed of P immediately after the collision,

the speed of Q immediately after the collision.

The points A, B and C lie in a horizontal plane. A batsman strikes a ball of mass 0.25 kg. Immediately before being struck, the ball is moving along the horizontal line AB with speed 30 m s^–1. Immediately after being struck, the ball moves along the horizontal line BC with speed 40 m s^–1. The line BC makes an angle of 60° with the original direction of motion AB, as shown in Figure 1. Find, to 3 significant figures,

the magnitude of the impulse given to the ball,

the size of the angle that the direction of this impulse makes with the original direction of motion AB.

A particle of mass 0.25 kg is moving with velocity (3i + 7j) m s^–1 when it receives the impulse (5 i – 3 j) N s. Find the speed of the particle immediately after the impulse.

A particle P of mass 0.5 kg is moving under the action of a single force F newtons. At time t seconds,

The velocity of P at time t seconds is v m s^–1. When t = 0, v = i – 4j.

Find v at time t seconds.

When t = 3, the particle P receives an impulse (–5 i + 12 j) N s.

Find the speed of P immediately after it receives the impulse.

At time t seconds (t 0), a particle P has position vector p metres, with respect to a fixed origin O, where

Find

the velocity of P at time t seconds,

the value of t when P is moving parallel to the vector i.

When t = 1, the particle P receives an impulse of (2 i – 6 j) N s. Given that the mass of P is 0.5 kg,

find the velocity of P immediately after the impulse.

A particle P of mass 2 m is moving with speed 2 u in a straight line on a smooth horizontal plane. A particle Q of mass 3m is moving with speed u in the same direction as P. The particles collide directly. The coefficient of restitution between P and Q is

Show that the speed of Q immediately after the collision is .

Find the total kinetic energy lost in the collision.

After the collision between P and Q, the particle Q collides directly with a particle R of mass m which is at rest on the plane. The coefficient of restitution between Q and R is e.

Calculate the range of values of e for which there will be a second collision between P and Q.

Two particles, P and Q, have masses 3 m and m respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. The speeds of P and Q immediately before the collision are 2 u and 4 u respectively. The magnitude of the impulse received by each particle in the

Find the speed of P after the collision.

Find the speed of Q after the collision.

A ball of mass 0.2 kg is projected vertically downwards with speed U m s^–1 from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of 10 m s^–1. The ball receives an impulse of magnitude 7 N s in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find

the value of U.

After hitting the ground, the ball moves vertically upwards and passes through a point B which is 1 m above the ground.

Find the time between the instant when the ball hits the ground and the instant when the ball first passes through B.

Sketch a velocity-time graph for the motion of the ball from when it was projected from A to when it first passes through B. (You need not make any further calculations to draw this sketch.)

Two particles, P and Q, have masses 2 m and 3 m respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of P is 4 u and the speed of Q is 3 u. As a result of the collision, Q has its direction of motion reversed and is moving with speed u.

Find the speed of P immediately after the collision.

State whether or not the direction of motion of P has been reversed by the collision.

Find the magnitude of the impulse exerted on P by Q in the collision.

A particle P of mass 0.4 kg is moving on rough horizontal ground when it hits a fixed vertical plane wall. Immediately before hitting the wall, P is moving with speed 4 m s^–1 in a direction perpendicular to the wall. The particle rebounds from the wall and comes to rest at a distance of 5 m from the wall. The coefficient of friction between P and the

Find the magnitude of the impulse exerted on P by the wall.

Figure 2

Two particles A and B have masses 2 m and 3 m respectively. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and A and B are above a horizontal plane, as shown in Figure 2. The system is released from rest.

Show that the tension in the string immediately after the particles are released

After descending 1.5 m, B strikes the plane and is immediately brought to rest. In the subsequent motion, A does not reach the pulley.

Find the distance travelled by A between the instant when B strikes the plane and the instant when the string next becomes taut.

Given that m = 0.5 kg,

find the magnitude of the impulse on B due to the impact with the plane.

Two particles A and B, of mass 2 kg and 3 kg respectively, are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately before the collision the speed of A is 5 m s^–1 and the speed of B is 6 m s^–1. The magnitude of the impulse exerted on B by A is 14 N s. Find

the speed of A immediately after the collision,

the speed of B immediately after the collision.

Particle P has mass 3 kg and particle Q has mass m kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of P is 4 m s^–1 and the speed of Q is 3 m s^–1. In the collision the direction of motion of P is unchanged and the direction of motion of Q is reversed. Immediately after the collision, the speed of P is 1 m s^–1 and the speed of Q is 1.5 m s^–1.

Find the magnitude of the impulse exerted on P in the collision.

Find the value of m.

Two particles P and Q have masses 4 m and m respectively. The particles are moving towards each other on a smooth horizontal plane and collide directly. The speeds of P and Q immediately before the collision are 2u and 5u respectively. Immediately after the collision, the speed of P is and its direction of motion is reversed.

Find the speed and direction of motion of Q after the collision.

Find the magnitude of the impulse exerted on P by Q in the collision.

Two particles A and B, of mass 5 m kg and 2 m kg respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of A and B are 3 m s^–1 and 4 m s^–1 respectively. The direction of motion of A is unchanged by the collision. Immediately after the collision, the speed of A is 0.8 m s^–1.

Find the speed of B immediately after the collision.

In the collision, the magnitude of the impulse exerted on A by B is 3.3 N s.

Find the value of m.

Particle P has mass 3 kg and particle Q has mass 2 kg. The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, P has speed 3 m s^–1 and Q has speed 2 m s^–1. Immediately after the collision, both particles move in the same direction and the difference in their speeds is 1 m s^–1.

Find the speed of each particle after the collision.

Find the magnitude of the impulse exerted on P by Q.

Two particles B and C have mass m kg and 3 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table. The two particles collide directly. Immediately before the collision, the speed of B is 4 m s¹ and the speed of C is 2 m s¹. In the collision the direction of motion of C is reversed and the direction of motion of B is unchanged. Immediately after the collision, the speed of B is 1 m s¹ and the speed of C is 3 m s¹. Find

the value of m,

the magnitude of the impulse received by C.

Particle P has mass m kg and particle Q has mass 3 kg. m The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision P has speed 4u m s¹ and Q has speed ku m s¹, where k is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved.

Find the value of k.

Find, in terms of m and u, the magnitude of the impulse exerted on P by Q.

A particle A of mass 2 kg is moving along a straight horizontal line with speed 12 m s^–1. Another particle B of mass m kg is moving along the same straight line, in the opposite direction to A, with speed 8 m s^–1. The particles collide. The direction of motion of A is unchanged by the collision. Immediately after the collision, A is moving with speed 3 m s^–1 and B is moving with speed 4 m s^–1. Find

the magnitude of the impulse exerted by B on A in the collision,

the value of m.

Two particles A and B are moving on a smooth horizontal plane. The mass of A is 2 m and the mass of B is m. The particles are moving along the same straight line but in opposite directions and they collide directly. Immediately before they collide the speed of A is 2 u and the speed of B is 3 u. The magnitude of the impulse received by each particle in the

Find

the speed of A immediately after the collision,

the speed of B immediately after the collision.

Two particles P and Q have mass 0.4 kg and 0.6 kg respectively. The particles are initially at rest on a smooth horizontal table. Particle P is given an impulse of magnitude 3 N s in the direction PQ.

Find the speed of P immediately before it collides with Q.

Immediately after the collision between P and Q, the speed of Q is 5 m s^–1.

Show that immediately after the collision P is at rest.

Two particles A and B have masses 4 kg and m kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of A is 5 m s^–1 and the speed of B is 3 m s^–1. Immediately after the collision, the direction of motion of A is unchanged and the speed of A is 1 m s^–1.

Find the magnitude of the impulse exerted on A in the collision.

Immediately after the collision, the speed of B is 2 m s^–1.

Find the value of m.