Momentum & Collisions (CIE A Level Maths: Mechanics)

Exam Questions

4 hours29 questions
1a
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4 marks

Three smooth spheres P comma space Q space and space R with masses 20 kg, 35 kg and m kg respectively, are travelling in the same direction along a straight line with speeds 7 space m space s to the power of negative 1 end exponent comma space u space m space s to the power of negative 1 end exponent space and space 10 space m space s to the power of negative 1 end exponent.

Given that P comma space Q space and space R all have the same momentum, find

(i)
the value of m,
(ii)
the value of u.
1b
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1 mark

Given that there are no collisions between any of P comma space Q space and space R state which sphere is travelling at the front, and which is travelling at the back.

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2a
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2 marks

A ball of mass 400 grams is travelling at a speed of 2 space m space s to the power of negative 1 end exponent. Calculate the momentum of the ball, giving your answer in Newton seconds.

2b
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2 marks

A car of mass m kg with a constant speed of 90 space k m space h to the power of negative 1 end exponent has momentum 32 500 N s. Find the value of m.

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3a
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3 marks

A particle P has mass 8 kg and a particle Q has mass 5 kg. P is moving towards Q in a straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision the speed of P is 3 m s-1 and Q is at rest.

Immediately after the collision P is travelling in the same direction with speed 1 m s-1.

Use the principle of conservation of linear momentum to find the speed of Q immediately after the collision.

3b
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2 marks

Show that the magnitude of the impulse exerted by Q on P in the collision is 16 N s.

3c
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2 marks

Verify that the change in momentum of Q due to the collision is also 16 space N space s but in the opposite direction.

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4a
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1 mark

Two particles, P and Q, with masses 10 kg and 16 kg respectively, are moving in the same direction along the same straight line when they collide directly. Immediately before they collide, P is moving with speed 13 space m space s to the power of negative 1 end exponent and Q is moving with speed 1 space m space s to the power of negative 1 end exponent. Immediately after they collide, Q is moving with speed 10 space m space s to the power of negative 1 end exponent.

Explain how you know the direction of motion of Q does not change after the collision.

4b
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4 marks
(i)
Find the speed of P after the collision.

(ii)
Explain how you know the direction of motion of P has reversed.

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5a
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3 marks

The masses of two particles A and B are 3 kg and m kg respectively. The particles are moving in a straight line, in opposite directions, on a smooth horizontal surface and collide directly.

Immediately before the collision the speed of A is 6 space m space s to the power of negative 1 end exponent and the speed of B is 5 space m space s to the power of negative 1 end exponent. As a result of the collision the direction of motion of A remains the same but the direction of motion of B is reversed. Immediately after the collision the speed of A is 4 space m space s to the power of negative 1 end exponent and the speed of B is 3 space m space s to the power of negative 1 end exponent

Show that m equals 0.75

5b
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3 marks

Find the change in momentum of A due to the collision.

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6a
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3 marks

Two friends, Safa and Adam, each roll a ball towards each other along the same straight line on a smooth horizontal table. Adam's ball has a mass of 0.8 kg.

Immediately before the collision, Safa's ball has speed 6 space m space s to the power of negative 1 end exponent and Adam's ball has speed 3 space m space s to the power of negative 1 end exponent.

Immediately after the collision, Safa's ball has speed 2 space m space s to the power of negative 1 end exponent and its direction of motion is reversed.

Immediately after the collision, the magnitude of the momentum of Safa's ball has decreased by 2.4 space k g space m space s to the power of negative 1 end exponent.

Show that the mass of Safa's ball is 0.6 kg.

6b
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4 marks
(i)
Find the speed of Adam's ball after the collision.

(ii)
State whether the direction of motion of Adam's ball is changed by the collision.

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7a
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2 marks

Two spheres A and B, with equal radii, are moving in the same direction along the same straight line on a smooth horizontal ground when they collide directly. The directions of motion of A and B are unchanged by the collision.

Immediately before the collision, A has speed 9 space m space s to the power of negative 1 end exponent and B has speed 1 space m space s to the power of negative 1 end exponent.

Immediately after the collision, A has speed 1.8 space m space s to the power of negative 1 end exponent and B has speed 5.8 space m space s to the power of negative 1 end exponent.

Given that the loss in the kinetic energy of A is 388.8 space straight J comma show that the mass of A is 10 space k g.

7b
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3 marks

Show that the mass of B is 15 space k g.

7c
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3 marks

After the collision, B continues to move with constant speed 5.8 space m space s to the power of negative 1 end exponent along the same straight line when it collides directly with another sphere C of equal radius with mass 30 space k g. Before the collision C is at rest and after the collision C is moving with speed 3 space m space s to the power of negative 1 end exponent.

(i)
Calculate the speed of B immediately after the collision with C.

(ii)
State whether the collision between B and C changes the direction of motion of B.
7d
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1 mark

Explain why there will be another collision between A and B.

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1a
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3 marks

Two objects A and B have masses 0.6 kg and 1.4 kg respectively. They are moving towards each other in opposite directions in a straight line on a smooth horizontal plane. The objects collide directly and coalesce to form an object C. Immediately before the collision, A is moving with speed 8 m s-1 and B is moving with speed 6 m s-1.

By modelling the objects as particles, find the speed and direction of motion of C after the collision.

1b
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1 mark

At the instance that object C is moving with speed u space m space s to the power of negative 1 end exponent, the magnitude of its momentum is 13 N s.

Find the value of u.

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2
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4 marks

Two marbles M and N have masses 3 kg and m kg respectively. The marbles are rolled towards each other in a straight line on a smooth horizontal table and they collide directly bringing M to rest. Immediately before the collision, the speeds of M and N are 3 space m space s to the power of negative 1 end exponent and 5 space m space s to the power of negative 1 end exponent respectively. Immediately after the collision, N moves with speed 4 space m space s to the power of negative 1 end exponent. The marbles are modelled as particles.

Find the value of m.

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3a
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3 marks

Two particles, P and Q, with masses 3 m and 5 m respectively are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. They collide directly. Immediately before the collision, the speed of P is 3 u and the speed of Q is 2 u. Immediately after the collision, the speed of P is halved and its direction of motion is reversed.

Find the speed of Q immediately after the collision.

3b
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1 mark

State whether the direction of motion of Q is changed immediately after the collision.

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4
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6 marks

Two small balls Aand B have masses m kg and 1.5 kg respectively. They are moving in the same direction along the same straight line on a smooth horizontal table when they collide directly. Immediately before they collide, Ais moving with speed 8 space m space s to the power of negative 1 end exponent and B is moving with speed 3 space m space s to the power of negative 1 end exponent. Immediately after they collide, A is moving with speed 1 space m space s to the power of negative 1 end exponent and B is moving with speed 4 space m space s to the power of negative 1 end exponent. The direction of motion of B is unchanged after the collision.

Find the value of m in the case that:

(i)
the direction of motion of A is unchanged after the collision.

(ii)
the direction of motion of A is reversed after the collision.

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5a
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3 marks

Amber works at a circus and routinely performs the human cannonball act. The act involves Amber, mass 65 kg, being ejected horizontally out of a specially designed cannon, mass 300 kg, into a net 100 metres away. Both the cannon and Amber start from rest before Amber is shot out at a speed of 108 km h-1.

(i)
Find the speed of the cannon after ejecting Amber.
(ii)
Show that the cannon recoils after ejecting Amber.
5b
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3 marks

Amber is travelling with a horizontal speed of 26 m s-1 when she hits the net. The horizontal resistance force is assumed to be constant.

Find the time it takes Amber to reach the net after being ejected from the canon.

5c
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4 marks

Calculate the magnitude of the resistance force.

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6a
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4 marks

Two spheres A and B are of equal radius and have masses 5 space k g and m space k g respectively. A and B are moving in the same direction in a straight line on a smooth horizontal surface when they collide directly. Immediately before the collision, A and B are moving with speeds 12 space m space s to the power of negative 1 end exponent and 2 space m space s to the power of negative 1 end exponent respectively. Immediately after the collision, the speed of A is 2 space m space s to the power of negative 1 end exponent, the speed of B is 4 space m space s to the power of negative 1 end exponent and the direction of motion of B is unchanged.

Find the two possible values of m.

6b
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3 marks

Show that the minimum total kinetic energy lost in the collision is 140 space straight J.

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7a
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3 marks

Three uniform spheres A comma space B and C of equal radius move in the same direction on the same straight line on a smooth horizontal table with B in the middle of A and C. The masses of A comma space B and C are 0.1 space k g comma space 0.3 space k g and 0.5 space k g respectively and A moves with speed 12 space m space s to the power of negative 1 end exponent while B and C are at rest. A and B collide directly which does not change the direction of motion of A and subsequently A moves with speed 1.5 space m space s to the power of negative 1 end exponent.

Show that the speed of B immediately after it collides with A is 3.5 space m space s to the power of negative 1 end exponent.

7b
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3 marks

In the subsequent motion, B collides directly with C. Immediately after this collision, C moves with speed 1.5 space m space s to the power of negative 1 end exponent.

Show that there will be another collision between A and B.

7c
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5 marks

Given that A and B coalesce in this collision,

show that the speed of the combined spheres is 1.125 space m space s to the power of negative 1 end exponent comma

show that the total loss of kinetic energy due to the three collisions is 6.38 space straight J correct to 3 significant figures.

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1a
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4 marks

Two objects P and Q have masses 0.6 kg and m kg respectively. They are moving towards each other in opposite directions in a straight line on a smooth horizontal surface. The objects collide directly and coalesce to form an object R. Immediately before the collision, P is travelling with speed 10 space m space s to the power of negative 1 end exponent and Q is travelling with speed 15 space m space s to the power of negative 1 end exponent. The speed of R after the collision is 12 space m space s to the power of negative 1 end exponent.

Find the value of m.

1b
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4 marks

There is a constant resistance force acting on R which causes R to come to rest after travelling 120 m.

Find the magnitude of the resistance force acting on R.

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2a
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4 marks

Two particles, A and B, with masses 7 m and 4 m respectively are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of Ais 3 uand the speed of B is 5 u. Immediately after the collision, the speed of Ais v subscript A and the speed of B is v subscript B.

The direction of motion of Ais reversed after the collision.

Show that v subscript B space end subscript greater than u over 4.

2b
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1 mark

State whether the direction of motion of B is reversed after the collision.

2c
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3 marks

Given that v subscript B is double v subscript A comma find the values of v subscript A and v subscript B.

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3a
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6 marks

Lucy and Fran are driving monster trucks along the same straight line in opposite directions towards each other. The combined mass of Lucy and her truck is 5400 kg and the combined mass of Fran and her truck is 4700 kg. They collide directly. Immediately before the collision, Lucy’s truck is moving with speed 4 space m space s to the power of negative 1 end exponent and Fran’s truck is moving with speed 5 space m space s to the power of negative 1 end exponent. Immediately after the collision, Lucy and Fran’s monster trucks are moving at the same speed.

(i)
Given that Lucy and Fran’s monster trucks are moving in the same direction after the collision, find the speed at which they both travel.
(ii)
State which person’s direction of motion has been reversed.
(iii)
Calculate the change in momentum of Fran’s monster truck due to the collision.
3b
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4 marks

Given instead that Lucy and Fran’s monster trucks bounce off each other and move in opposite directions with equal speeds after the collision, calculate the speed at which they both travel.

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4a
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6 marks

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A ball of mass 2 kg is projected along a rough horizontal floor in a straight line with speed 8 space m space s to the power of negative 1 end exponent. Initially the ball is 20 m away from the wall as shown in the diagram above. The coefficient of friction between the ball and the floor is 0.1.

Find the momentum of the ball immediately before it hits the wall.

4b
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3 marks

The ball rebounds from the wall and travels for 3 space s before coming to rest.

Find the momentum of the ball immediately after it hits the wall.

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5a
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4 marks

Two particles P and Q, with masses 0.4 kg and 2.4 kg respectively, are moving in the same direction along a straight line on a smooth horizontal plane. At the instant when P has speed 10 space m space s to the power of negative 1 end exponent and Q has speed u space m space s to the power of negative 1 end exponent, the two particles collide directly. Immediately after the collision, the two particles are moving in opposite directions and the speed of P is half of the speed of Q.

Show that the speed of Q immediately after the collision is 2 over 11 space left parenthesis 10 plus 6 u right parenthesis space m space s to the power of negative 1 end exponent

5b
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5 marks

After the collision Q continues travelling in the same straight line for 3 seconds before being brought to rest by a constant resistive force of magnitude 3.2 N.

Calculate the value of u.

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6
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4 marks

Paul, a teenager of mass 50 space k g, is skating in a straight line on a skateboard of mass 5 space k g. In the instant that Paul and the skateboard are travelling at 3 space m space s to the power of negative 1 end exponent, Paul jumps forward off the skateboard. Paul’s speed is 4 space m space s to the power of negative 1 end exponent immediately after he jumps off the skateboard.

(i)
Calculate the speed of the skateboard immediately after Paul jumps off.
(ii)
State whether the skateboard continues to move in the same direction as Paul.

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7a
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4 marks

Three small balls A comma space B and C of equal radius have masses 0.2 space k g comma 0.3 space k g and 1.5 space k g respectively. The three balls lie in the same straight line on a smooth horizontal ground with B in the middle of A and C. A and B are projected towards each other in opposite directions and collide directly when they are travelling with speeds 11 space m space s to the power of negative 1 end exponent and 2 space m space s to the power of negative 1 end exponent respectively. Immediately after the collision, the speed of B is twice the speed of A.

Find the two possible values for the speed of B immediately after the collision.

7b
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3 marks

Immediately after the collision the speed of B is the larger of the two values found in part (a). In the subsequent motion, B collides with C which is at rest. After the collision, C moves with speed v space m space s to the power of negative 1 end exponent.

Given that there are no further collisions between A and B, find the maximum value of v.

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1
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6 marks

Two objects A and B have masses 10 m and 7 m respectively. The objects are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The objects collide directly which causes the direction of motion of A to be reversed.

Immediately before the objects collide, the speed of A is 5 u and the speed of B is lambda u, where lambda is a constant.

Immediately after the objects collide, the speed of A is 3 u and the speed of B is 2 u.

Find the two possible values of lambda.

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2a
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4 marks

Two particles P and Q move towards each other in opposite directions along a straight line on a smooth horizontal surface. The mass of P is 9 m and the mass of Q is 14 m. They collide directly. Immediately before the collision, P moves with speed 16 u and Q moves with speed 7 u. Immediately after the collision the directions of P and Q are reversed and P has speed v subscript p and Q has speed v subscript Q.

Show that v subscript Q space greater than space 23 over 7 u.

2b
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6 marks

After the collision, both particles decelerate at a constant rate of 2 space straight m s to the power of negative 2 end exponent. Particle P takes 0.3 u seconds longer to come to rest than particle Q.

Find the values of v subscript p and v subscript Q.

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3
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6 marks

Harry drops a cricket ball, mass 0.16 kg, from a height 2 m above the ground. The only force acting on the cricket ball as it falls is the force due to gravity. The ball falls in a straight vertical line, hits the ground and bounces back upwards still following the same straight line. Immediately after the ball hits the ground, the magnitude of its momentum decreases by 0.584 space kg space straight m space straight s to the power of negative 1 end exponent.

Find the greatest height above the ground reached by the cricket ball after it bounces off the ground.

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4
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5 marks

An empty car of mass m kg is stopped in the middle of a road. A lorry of mass 3800 kg is driving at a constant speed of 48 space km space straight h to the power of negative 1 end exponent when it collides with the car. The magnitude of the momentum of the car immediately after the collision is 12500 space kg space straight m space straight s to the power of negative 1 end exponent.

Given that the speeds of the lorry and the car are equal immediately after the collision, find their common speed and find the value of m.

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5a
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7 marks

In this question, take g space equals space 10 space straight m space straight s to the power of negative 2 end exponent.

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AB is a line of greatest slope, of length 4 m, on a smooth plane inclined at 40 degree to the horizontal. Particle P of mass 0.4 kg is projected down the slope from the point A and particle Q of mass 0.33 kg is projected up the slope from the point B. When P has speed 1 space straight m space straight s to the power of negative 1 end exponent and Q is moving up the slope with speed 5 space straight m space straight s to the power of negative 1 end exponent, the particles collide directly at a point 1 m down the slope from A, as shown in the diagram above. Immediately after the collision, the direction of motion of Q is reversed and it has speed 1 space straight m space straight s to the power of negative 1 end exponent.

Show that P reaches the top of the slope, A, in the subsequent motion.

5b
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7 marks

Determine, with calculations, whether P reaches A before Q reaches B.

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6a
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6 marks

Armina and Dillon are two armadillos with masses 5 kg and m kg respectively. They curl into balls and roll towards each other in opposite directions along a straight line on smooth ground. They collide directly which causes Armina's direction of motion to be reversed. Immediately before the collision the speed of Armina is 3 space straight m space straight s to the power of negative 1 end exponent and the speed of Dillon is 6 space straight m space straight s to the power of negative 1 end exponent. Immediately after the collision, the speed of Armina is 2.6 space straight m space straight s to the power of negative 1 end exponent and the speed of Dillon is 2 space straight m space straight s to the power of negative 1 end exponent. Armina and Dillon are modelled as particles.

Find the two possible values for the loss of kinetic energy in the collision.

6b
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6 marks

After the collision, Armina continues to roll with a constant speed until she reaches some rough ground. Once Armina enters the rough ground she continues to roll for 2 seconds before coming to rest

Find the coefficient of friction between Armina and the rough ground.

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7
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7 marks

A particle P of mass 5 kg is travelling along a straight line on a smooth horizontal surface with constant speed 3 space straight m space straight s to the power of negative 1 end exponent. It collides directly with particle Q of mass 2 kg which is moving along the same straight line. Immediately after the collision, the direction of motion of P is reversed and its speed is 1 space straight m space straight s to the power of negative 1 end exponent. The kinetic energy lost in the collision is 60 J.

(i)
Calculate the speed of Q immediately before the collision. Clearly state the direction of motion.
(ii)
Calculate the speed of Q immediately after the collision. Clearly state the direction of motion.

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8
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6 marks

R, S and T are three small marbles of equal radius and masses 0.4 kg, 0.2 kg and 0.1 kg respectively. The three marbles lie in the same straight line on a smooth horizontal floor with S in the middle. R and S are projected towards each other with speeds 3 space straight m space straight s to the power of negative 1 end exponent and 1 space straight m space straight s to the power of negative 1 end exponent respectively and collide directly. At the instant that R and S collide, T is projected towards S with a speed of 1 space straight m space straight s to the power of negative 1 end exponent. In the subsequent motion, R travels towards T with a speed of v space straight m space straight s to the power of negative 1 end exponent and S travels towards T with a speed of 2 v space straight m space straight s to the power of negative 1 end exponent. S and T collide directly and T rebounds with speed x space straight m space straight s to the power of negative 1 end exponent.

Given that there are no further collisions, find the maximum value of x.

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