Direct impact & restitution

Three small balls, A, B and C, have masses 2m, 3m and 4m respectively.

The balls are initially at rest in a straight line on a smooth horizontal surface, with B between A and C, as shown in Figure 2.

Ball A is projected towards B with speed 5u and A and B collide directly. Immediately after the collision, the speed of B is 3 u

The balls are modelled as uniform spheres with equal radii.

Find, in terms of u, the speed of A immediately after the collision with B.

Find the coefficient of restitution between A and B.

After the collision between A and B, ball C is projected towards B with speed u Balls B and C collide directly. The coefficient of restitution between B and C is f

Given that there is a second direct collision between A and B

find the complete range of possible values of f

A particle A of mass 2m is moving in a straight line with speed 3u on a smooth horizontal plane. Particle A collides directly with a particle B of mass m which is at rest on the plane.

The coefficient of restitution between A and B is e, where e 0

Show that the speed of B immediately after the collision is 2u(1 + e).

After the collision, B hits a smooth fixed vertical wall which is perpendicular to the direction of motion of B.

Show that there will be a second collision between A and B.

The coefficient of restitution between B and the wall is

Find, in simplified form, in terms of m, u and e,

the magnitude of the impulse received by B in its collision with the wall,

the loss in kinetic energy of B due to its collision with the wall.

A particle P of mass 2 m is moving in a straight line with speed 3 u on a smooth horizontal plane. It collides directly with a particle Q of mass m that is moving on the plane with speed 2 u in the opposite direction to P.

The coefficient of restitution between P and Q is e, where e >

After the collision Q hits a smooth fixed vertical wall that is perpendicular to the direction of motion of Q. The coefficient of restitution between Q and the wall is f.

Find, in terms of e, the set of values of f for which there will be a second collision between P and Q.

Two particles, A and B, are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.

Particle A has mass 5m and particle B has mass 3m.

The coefficient of restitution between A and B is e, where e 0

Immediately after the collision the speed of A is v and the speed of B is 2v.

Given that A and B are moving in the same direction after the collision,

find the set of possible values of e.

Given also that the kinetic energy of A immediately after the collision is 16% of the kinetic energy of A immediately before the collision,

find

the value of e,

the magnitude of the impulse received by A in the collision, giving your answer in terms of m and v.

Two particles, A and B, have masses 3m and 4m respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of A is 2 u and the speed of B is u.

The coefficient of restitution between A and B is e.

Show that the direction of motion of each of the particles is unchanged by the collision.

After the collision with A, particle B collides directly with a third particle, C, of mass 2m, which is at rest on the surface.

The coefficient of restitution between B and C is also e.

Show that there will be a second collision between A and B.