A LevelFurther Maths: Further Mechanics 1EdexcelTopic QuestionsElastic Collisions in 2DElastic Collisions in 2D

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

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1a
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Figure 1

Figure 1 represents the plan view of part of a horizontal floor, where A B and B C are perpendicular vertical walls.

The floor and the walls are modelled as smooth.

A ball is projected along the floor towards A B with speed u ms−1 on a path at an angle of 60° to A B. The ball hits A B and then hits B C.

The ball is modelled as a particle.
The coefficient of restitution between the ball and wall A B is fraction numerator 1 over denominator square root of 3 end fraction.
The coefficient of restitution between the ball and wall B C is square root of 2 over 5 end root.
Show that, using this model, the final kinetic energy of the ball is 35% of the initial kinetic energy of the ball.

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1b
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In reality the floor and the walls may not be smooth. What effect will the model have had on the calculation of the percentage of kinetic energy remaining?

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2a
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[In this question bold i and bold j 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 open parentheses 3 bold i plus 3 bold j close parentheses space ms to the power of negative 1 end exponent and the velocity of B is open parentheses negative 5 bold i plus 2 bold j close parentheses space ms to the power of negative 1 end exponent.

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

The coefficient of restitution between the spheres is 1 fourth.

Find the velocity of B immediately after the collision.

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2b
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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.

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3a
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fig-2-june-2019-9fm0-a-level-further-maths

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 alpha with AB, where tan space alpha equals 4 over 3. The ball hits AB and then hits BC.

Immediately after hitting AB, the ball is moving at an angle beta to AB, where tan space beta equals 1 third.

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

The coefficient of restitution between the ball and BC is 1 half.

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

show that the value e space equals space 1 fourth.

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3b
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Find the speed of the ball immediately after it hits BC.

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3c
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Suggest two ways in which the model could be refined to make it more realistic.

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4a
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[In this question bold i and bold j 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 open parentheses 3 bold i plus 2 bold j close parentheses space ms to the power of negative 1 end exponent and the velocity of B is open parentheses negative 4 bold i minus bold j close parentheses space ms to the power of negative 1 end exponent.

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

The coefficient of restitution between the spheres is 3 over 7.

Find the velocity of A immediately after the collision. 

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4b
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Find the magnitude of the impulse received by A in the collision. 

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4c
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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. 

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5a
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[In this question, bold i and bold j are perpendicular unit vectors in a horizontal plane.]

fig-1-nov-2020-9fm0-3c-further-mechanics-edexcel

Figure 1

Figure 1 represents the plan view of part of a smooth horizontal floor, where A B 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 left parenthesis 7 bold i space plus space 2 bold j right parenthesis space ms to the power of negative 1 end exponent.

Immediately after the impact the velocity of the ball is left parenthesis bold i space plus space 6 bold j right parenthesis space ms to the power of negative 1 end exponent.

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

Show that A B is parallel to left parenthesis 2 bold i space plus space 3 bold j right parenthesis.

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5b
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Find the value of e.

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6a
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A smooth uniform sphere P has mass 0.3 kg. Another smooth uniform sphere Q, with the same radius asspace P, 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 P is left parenthesis 4 bold i bold space plus space 2 bold j right parenthesis space ms to the power of negative 1 end exponent and the velocity of Q is left parenthesis – 3 bold i space plus space bold j right parenthesis space ms to the power of negative 1 end exponent.

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

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

Find

i)
the velocity of P immediately after the collision,
ii)
the velocity of Q immediately after the collision,
iii)
the coefficient of restitution between P and Q,  
carefully justifying your answers.

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6b
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Find the size of the angle through which the direction of motion of P is deflected by the collision.

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7a
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fig-2-nov-2020-9fm0-3c-further-mechanics-edexcel

Figure 2

Figure 2 represents the plan view of part of a horizontal floor, where A B and C D represent fixed vertical walls, with A B spaceparallel to C D.

A small ball is projected along the floor towards wall A B. Immediately before hitting wall A B, the ball is moving with speed v ms–1 at an angle alpha to A B space, where 0 space less than space alpha space less than space pi over 2.

The ball hits wall A B spaceand then hits wall space C D.

After the impact with wall space C D, the ball is moving at angle 1 half alpha tospace C D.

The coefficient of restitution between the ball and wall A B spaceis 2 over 3.

The coefficient of restitution between the ball and wall C D is also 2 over 3.

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

Show that tan open parentheses 1 half alpha close parentheses equals 1 third.

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7b
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Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.

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8a
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[In this question, bold i and bold j are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere P has mass 0.3 kg. Another smooth uniform sphere Q, with the same radius as P, 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 P is left parenthesis u bold i space plus space 2 bold j right parenthesis space ms to the power of negative 1 end exponent, where u is a positive constant, and the velocity of Q is open parentheses negative 4 bold i space plus space 3 bold j close parentheses space ms to the power of negative 1 end exponent.

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

The coefficient of restitution between P and Q is 3 over 5.

As a result of the collision, the direction of motion of P is deflected through an angle of 90° and the direction of motion of Q is deflected through an angle of alpha degree space

Find the value of u.

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8b
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Find the value of alpha.

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8c
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State how you have used the fact that P and Q have equal radii.

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9a
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fig-1-nov-2021-9fm0-3c-further-mechanics-edexcel

Figure 1

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

A small ball is projected along the floor towards the wall A B. Immediately before hitting the wall A B the ball is moving with speed v space ms to the power of negative 1 end exponent at an angle theta to A B.

The ball hits the wall A B and then hits the wall B C.
The coefficient of restitution between the ball and the wall A B is 1 third.

The coefficient of restitution between the ball and the wall B C is e.

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 A B. 

Find the exact value of cos space theta.

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9b
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The ball loses half of its remaining kinetic energy in the impact with the wall B C.

Find the exact value of e.

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10a
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[In this question, bold i and bold j are perpendicular unit vectors in a horizontal plane.]

fig-3-nov-2021-9fm0-3c-further-mechanics-edexcel

Figure 3

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

The direction of stack A B with rightwards arrow on top is in the direction of the vector left parenthesis bold i space plus space bold j right parenthesis.

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

Immediately before its impact with the wall A B, the velocity of the ball is space left parenthesis 8 bold i space plus space 2 bold j right parenthesis space ms to the power of negative 1 end exponent.

Immediately after its impact with the wall A B, the velocity of the ball is bold v bold space ms to the power of negative 1 end exponent.

The coefficient of restitution between the ball and the wall is  1 third.

By modelling the ball as a particle,

show that bold v space equals space open parentheses 4 bold i space plus space 6 bold j close parentheses.

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10b
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Find the magnitude of the impulse received by the ball in the impact.

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