Elastic Collisions in 1D (Edexcel A Level Further Maths: Further Mechanics 1)

A particle P of mass 2m and a particle Q of mass 5m are moving along the same straight line on a smooth horizontal plane.

They are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of P is 2u and the speed of Q is u.

The direction of motion of Q is reversed by the collision.

The coefficient of restitution between P and Q is e. 

Find the range of possible values of e.

Given that 

show that the kinetic energy lost in the collision is .

Without doing any further calculation, state how the amount of kinetic energy lost in the collision would change if .

A particle P of mass 3m is moving in a straight line on a smooth horizontal floor. A particle Q of mass 5m is moving in the opposite direction to P along the same straight line.

The particles collide directly.

Immediately before the collision, the speed of P is 2u and the speed of Q is u.
The coefficient of restitution between P and Q is e.

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

Find the range of values of e for which the direction of motion of P is not changed as a result of the collision.

When P and Q collide they are at a distance d from a smooth fixed vertical wall, which is perpendicular to their direction of motion. After the collision with P, particle Q collides directly with the wall and rebounds so that there is a second collision between P and Q.
This second collision takes place at a distance x from the wall.

Given that  and the coefficient of restitution between Q and the wall is ,

find x in terms of d.

Two particles, A and B, of masses 2m and 3m respectively, are moving on a smooth horizontal plane. The particles are moving in opposite directions towards each other along the same straight line when they collide directly. Immediately before the collision the speed of A is 2u and the speed of B is u. In the collision the impulse of A on B has magnitude 5mu.

Find the coefficient of restitution between A and B.

Find the total loss in kinetic energy due to the collision.

Three particles, P, Q and R, are at rest on a smooth horizontal plane. The particles lie along a straight line with Q between P and R. The particles Q and R have masses m and km respectively, where k is a constant.

Particle Q is projected towards R with speed u and the particles collide directly.

The coefficient of restitution between each pair of particles is e.

Find, in terms of e, the range of values of k for which there is a second collision.

Given that the mass of P is km and that there is a second collision,

write down, in terms of u, k and e, the speed of Q after this second collision.

Figure 1

Figure 1 represents the plan of part of a smooth horizontal floor, where W₁ and W₂ are two fixed parallel vertical walls. The walls are 3 metres apart.

A particle lies at rest at a point O on the floor between the two walls, where the point O is d metres, , from W₁.

At time , the particle is projected from O towards W₁ with speed u ms^–1 in a direction perpendicular to the walls.
The coefficient of restitution between the particle and each wall is .
The particle returns to O at time t = T seconds, having bounced off each wall once.

Show that .

The value of u is fixed, the particle still hits each wall once but the value of d can now vary.

Find the least possible value of T, giving your answer in terms of u. You must give a reason for your answer. 

A particle P of mass 3m and a particle Q of mass 2m are moving along the same straight line on a smooth horizontal plane. The particles are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of P is u and the speed of Q is 2u.

Immediately after the collision P and Q are moving in opposite directions.

The coefficient of restitution between P and Q is e.

Find the range of possible values of e, justifying your answer.

Given that Q loses 75% of its kinetic energy as a result of the collision,

find the value of e.

Two particles and have masses and 4 respectively. The particles are at rest on a smooth horizontal plane. Particle is given a horizontal impulse, of magnitude , in the direction . Particle then collides directly with . Immediately after this collision, is at rest and has speed . The coefficient of restitution between the particles is .

Find in terms of and .

Show that .

Find, in terms of and , the total kinetic energy lost in the collision between and
.

Three particles  and are at rest on a smooth horizontal plane. The particles lie along a straight line with between  and .

Particle  has mass 4 and particle has mass , where is a positive constant.

Particle is projected with speed along the plane towards and they collide directly.

The coefficient of restitution between and is .

Find the range of values of for which there would be no further collisions.

The magnitude of the impulse on in the collision between and is .

Find the value of . 

A small ball, of mass , is thrown vertically upwards with speed  from a point on a smooth horizontal floor. The ball moves towards a smooth horizontal ceiling that is a vertical distance above . The coefficient of restitution between the ball and the ceiling is  .

In a model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to air resistance of constant magnitude .

Using this model,

use the work-energy principle to find, in terms of and , the speed of the ball immediately before it strikes the ceiling.

Find, in terms of and , the speed of the ball immediately before it strikes the floor at for the first time.

In a simplified model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to no air resistance.

Using this simplified model,

explain, without any detailed calculation, why the speed of the ball, immediately before it strikes the floor at for the first time, would still be less than .

Two particles, and , have masses 3 and 4 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 is 2 and the speed of is .

The coefficient of restitution between and is .

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

After the collision with , particle collides directly with a third particle, , of mass
, which is at rest on the surface.

The coefficient of restitution between and is also .

Show that there will be a second collision between and .

Figure 2

A particle of mass  is at rest on a smooth horizontal plane between two smooth fixed parallel vertical walls, as shown in the plan view in Figure 2. The particle is projected along the plane with speed  towards one of the walls and strikes the wall at right angles. The coefficient of restitution between the particle and each wall is and air resistance is modelled as being negligible.

Using the model,

find, in terms of  and , an expression for the total loss in the kinetic energy of the particle as a result of the first two impacts.

Given that  can vary such that  and using the model,

find the value of for which the total loss in the kinetic energy of the particle as a result of the first two impacts is a maximum.

Describe the subsequent motion of the particle.

Two particles,  and , have masses and respectively. The particles are moving on a smooth horizontal plane in the same direction along the same straight line when they collide directly. The coefficient of restitution between and is , where .

Immediately before the collision the speed of is  and the speed of is .

Show that the speed of immediately after the collision is .

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

The magnitude of the impulse on  in the collision is .

Find the possible values of .

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

Particle has mass 5 and particle has mass .

The coefficient of restitution between and is , where 

Immediately after the collision the speed of is  and the speed of is .

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

find the set of possible values of .

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

find

the value of ,

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