Elastic Collisions in 2D (Edexcel A Level Further Maths: Further Mechanics 1): Exam Questions

[In this question  and  are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere A has mass 2m kg and another smooth uniform sphere B, with the same radius as A, has mass 3m kg.

The spheres are moving on a smooth horizontal plane when they collide obliquely.

Immediately before the collision the velocity of A is and the velocity of B is .

At the instant of collision, the line joining the centres of the spheres is parallel to .

The coefficient of restitution between the spheres is .

Find the velocity of B immediately after the collision.

Find, to the nearest degree, the size of the angle through which the direction of motion of B is deflected as a result of the collision.

Figure 2

Figure 2 represents the plan view of part of a horizontal floor, where AB and BC are fixed vertical walls with AB perpendicular to BC.

A small ball is projected along the floor towards AB with speed 6 ms^–1 on a path that makes an angle with AB, where . The ball hits AB and then hits BC.

Immediately after hitting AB, the ball is moving at an angle to AB, where .

The coefficient of restitution between the ball and AB is e.

The coefficient of restitution between the ball and BC is .

By modelling the ball as a particle and the floor and walls as being smooth,

show that the value .

Find the speed of the ball immediately after it hits BC.

Suggest two ways in which the model could be refined to make it more realistic.

[In this question  and  are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere A has mass 0.2 kg and another smooth uniform sphere B, with the same radius as A, has mass 0.4 kg.

The spheres are moving on a smooth horizontal surface when they collide obliquely.
Immediately before the collision, the velocity of A is and the velocity of B is .

At the instant of collision, the line joining the centres of the spheres is parallel to .

The coefficient of restitution between the spheres is .

Find the velocity of A immediately after the collision. 

Find the magnitude of the impulse received by A in the collision. 

Find, to the nearest degree, the size of the angle through which the direction of motion of A is deflected as a result of the collision. 

[In this question, and are perpendicular unit vectors in a horizontal plane.]

Figure 1

Figure 1 represents the plan view of part of a smooth horizontal floor, where  represents a fixed smooth vertical wall.

A small ball of mass 0.5 kg is moving on the floor when it strikes the wall.

Immediately before the impact the velocity of the ball is .

Immediately after the impact the velocity of the ball is .

The coefficient of restitution between the ball and the wall is .

Show that  is parallel to .

Find the value of .

A smooth uniform sphere has mass 0.3 kg. Another smooth uniform sphere , with the same radius as, has mass 0.2 kg.

The spheres are moving on a smooth horizontal surface when they collide obliquely.
Immediately before the collision the velocity of is  and the velocity of is .

At the instant of collision, the line joining the centres of the spheres is parallel to .

The kinetic energy of immediately after the collision is half the kinetic energy of immediately before the collision.

Find

i) the velocity of immediately after the collision,

ii) the velocity of immediately after the collision,

iii) the coefficient of restitution between and ,  

carefully justifying your answers.

Find the size of the angle through which the direction of motion of is deflected by the collision.

Figure 2

Figure 2 represents the plan view of part of a horizontal floor, where and represent fixed vertical walls, with parallel to .

A small ball is projected along the floor towards wall . Immediately before hitting wall , the ball is moving with speed  ms^–1 at an angle  to , where .

The ball hits wall and then hits wall .

After the impact with wall , the ball is moving at angle  to.

The coefficient of restitution between the ball and wall is .

The coefficient of restitution between the ball and wall is also .

The floor and the walls are modelled as being smooth. The ball is modelled as a particle. 

Show that .

Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.

[In this question, and are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere  has mass 0.3 kg. Another smooth uniform sphere , with the same radius as , has mass 0.5 kg.

The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of is , where  is a positive constant, and the velocity of is .

At the instant when the spheres collide, the line joining their centres is parallel to .

The coefficient of restitution between and is .

As a result of the collision, the direction of motion of is deflected through an angle of 90° and the direction of motion of is deflected through an angle of 

Find the value of .

Find the value of .

State how you have used the fact that  and have equal radii.

Figure 1

Figure 1 represents the plan view of part of a horizontal floor, where and represent fixed vertical walls, with perpendicular to .

A small ball is projected along the floor towards the wall . Immediately before hitting the wall the ball is moving with speed  at an angle  to .

The ball hits the wall and then hits the wall . The coefficient of restitution between the ball and the wall is .

The coefficient of restitution between the ball and the wall is .

The floor and the walls are modelled as being smooth.

The ball is modelled as a particle.

The ball loses half of its kinetic energy in the impact with the wall . 

Find the exact value of .

The ball loses half of its remaining kinetic energy in the impact with the wall .

Find the exact value of .

[In this question, and are perpendicular unit vectors in a horizontal plane.]

Figure 3

Figure 3 represents the plan view of part of a smooth horizontal floor, where is a fixed smooth vertical wall. 

The direction of  is in the direction of the vector .

A small ball of mass 0.25 kg is moving on the floor when it strikes the wall .

Immediately before its impact with the wall , the velocity of the ball is .

Immediately after its impact with the wall , the velocity of the ball is .

The coefficient of restitution between the ball and the wall is  .

By modelling the ball as a particle,

show that .

Find the magnitude of the impulse received by the ball in the impact.

Two smooth uniform spheres, and , have equal radii. The mass of is and the mass of is . The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, is moving with speed at 30° to the line of centres of the spheres and is moving with speed at 30° to the line of centres of the spheres. The direction of motion of is turned through an angle of 90° by the collision, as shown in Figure 3.

(i) Find the size of the angle through which the direction of motion of is turned as a result of the collision.

(ii) Find, in terms of and , the magnitude of the impulse received by in the collision.

Figure 5 represents the plan view of part of a smooth horizontal floor, where and are smooth fixed vertical walls. The vector is in the direction of and the vector is in the direction of .

A small ball is projected across the floor towards . Immediately before the impact with , the velocity of is . The ball bounces off and then hits .

The ball is modelled as a particle.

Given that the coefficient of restitution between and is , find the full range of possible values of .

It is now given that and that the coefficient of restitution between and is .

Find, in terms of and , the velocity of immediately after its impact with .

A smooth uniform sphere of mass is moving with speed on a smooth horizontal plane. The sphere collides obliquely with another uniform sphere of mass which is at rest on the plane. The two spheres have the same radius.

Immediately before the collision the direction of motion of makes an angle , where , with the line joining the centres of the spheres.

Immediately after the collision the direction of motion of makes an angle with the line joining the centres of the spheres, as shown in Figure 1.

The coefficient of restitution between the spheres is .

Show that .

Given that , show that the directions of motion of the two spheres immediately after the collision are perpendicular.

A particle of mass is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle , where

At the instant immediately before the impact, the speed of is .

At the instant immediately after the impact, is moving horizontally with speed .

Show that the magnitude of the impulse exerted on the plane by is

The coefficient of restitution between and the plane is , where

Show that

Show that the kinetic energy lost by in the impact is

Hence find, in terms of , and only, the kinetic energy lost by in the impact.

A small smooth snooker ball is projected from the corner of a horizontal rectangular snooker table .

The ball is projected so it first hits the side at the point , then hits the side at the point and then returns to .

Angle , Angle , Angle .
The ball moves along with speed , along with speed and along with speed , as shown in Figure 2.

The coefficient of restitution between the ball and side is .

The coefficient of restitution between the ball and side is .

The ball is modelled as a particle.

Use the model to answer all parts of this question.

Show that

Hence show that

By considering (angle + angle ) or otherwise, show that it would be possible for the ball to return to only if

If instead , the ball would not return to .

Given that , use the result from part (b) to describe the path of the ball after it hits at , explaining your answer.

[In this question, and are horizontal perpendicular unit vectors.]

A particle is moving with velocity on a smooth horizontal plane.
The particle collides with a smooth vertical wall and rebounds with velocity

The coefficient of restitution between and the wall is .

Find the value of .

After the collision, goes on to hit a second smooth vertical wall, which is parallel to .

The coefficient of restitution between and this second wall is

The angle through which the direction of motion of has been deflected by its collision with this second wall is .

Find the value of , giving your answer to the nearest whole number.

A smooth uniform sphere of mass is moving with speed on a smooth horizontal plane. The sphere collides obliquely with a smooth uniform sphere of mass 3 which is at rest on the plane. The two spheres have the same radius.

Immediately before the collision, the direction of motion of makes an angle , where , with the line joining the centres of the spheres.

Immediately after the collision, the direction of motion of is perpendicular to its original direction, as shown in Figure 1.

The coefficient of restitution between the spheres is .

Show that the speed of immediately after the collision is

Show that

Show that