Direct impact & restitution

A particle A of mass 3 m is moving in a straight line with speed 2 u on a smooth horizontal floor. Particle A collides directly with another particle B of mass 2 m which is moving along the same straight line with speed u but in the opposite direction to A. The coefficient

Show that the speed of B immediately after the collision is

Find the speed of A immediately after the collision.

After the collision, B hits a smooth vertical wall which is perpendicular to the direction of motion of B. The coefficient of restitution between B and the wall is . The first collision between A and B occurred at a distance x from the wall. The particles collide again at a distance y from the wall.

Find y in terms of x.

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.

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 particle of mass m kg lies on a smooth horizontal surface. Initially the particle is at rest at a point O midway between a pair of fixed parallel vertical walls. The walls are 2 m apart. At time t = 0 the particle is projected from O with speed u m s^–1 in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is . The magnitude of the impulse on the particle due to the first impact with a wall is N s.

Find the value of .

The particle returns to O, having bounced off each wall once, at time t = 3 seconds.

Find the value of u.

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.

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 particle A of mass m is moving with speed u on a smooth horizontal floor when it collides directly with another particle B, of mass 3 m, which is at rest on the floor. The coefficient of restitution between the particles is e. The direction of motion of A is reversed by the collision.

Find, in terms of e and u,

the speed of A immediately after the collision,

the speed of B immediately after the collision.

After being struck by A the particle B collides directly with another particle C, of mass 4 m, which is at rest on the floor. The coefficient of restitution between B and C is 2 e. Given that the direction of motion of B is reversed by this collision,

find the range of possible values of e,

determine whether there will be a second collision between A and B.

A particle P of mass 3 m is moving with speed 2 u in a straight line on a smooth horizontal plane. The particle P collides directly with a particle Q of mass 4 m moving on the plane with speed u in the opposite direction to P. The coefficient of restitution between P and Q is e.

Find the speed of Q immediately after the collision.

Given that the direction of motion of P is reversed by the collision,

find the range of possible values of e.

A particle P of mass m is moving in a straight line on a smooth horizontal surface with speed 4 u. The particle P collides directly with a particle Q of mass 3 m which is at rest on the surface. The coefficient of restitution between P and Q is e. The direction of motion of P is reversed by the collision.

A particle P of mass m kg is moving with speed 6 m s^–1 in a straight line on a smooth horizontal floor. The particle strikes a fixed smooth vertical wall at right angles and rebounds. The kinetic energy lost in the impact is 64 J. The coefficient of restitution between P and the wall is

Show that m = 4.

After rebounding from the wall, P collides directly with a particle Q which is moving towards P with speed 3 m s^–1. The mass of Q is 2 kg and the coefficient of restitution

Show that there will be a second collision between P and the wall.

A small ball A of mass 3 m is moving with speed u in a straight line on a smooth horizontal table. The ball collides directly with another small ball B of mass m moving with speed u towards A along the same straight line. The coefficient of restitution between A and B is The balls have the same radius and can be modelled as particles.

Find

the speed of A immediately after the collision,

the speed of B immediately after the collision.

After the collision B hits a smooth vertical wall which is perpendicular to the direction of motion of B. The coefficient of restitution between B and the wall is

Find the speed of B immediately after hitting the wall.

The first collision between A and B occurred at a distance 4 a from the wall. The balls collide again T seconds after the first collision.

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.

Particles A, B and C of masses 4 m, 3 m and m respectively, lie at rest in a straight line on a smooth horizontal plane with B between A and C. Particles A and B are projected towards each other with speeds u m s^–1 and v m s^–1 respectively, and collide directly. As a result of the collision, A is brought to rest and B rebounds with speed k v m s^–1. The coefficient of restitution between A and B is

Show that u = 3 v.

Find the value of k.

Immediately after the collision between A and B, particle C is projected with speed 2v m s^–1 towards B so that B and C collide directly.

Show that there is no further collision between A and B.

A particle A of mass 4 m is moving with speed 3 u in a straight line on a smooth horizontal table. The particle A collides directly with a particle B of mass 3 m moving with speed 2 u in the same direction as A. The coefficient of restitution between A and B is e. Immediately after the collision the speed of B is 4 eu.

Find the total kinetic energy lost in the collision.

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.