Particle P has mass m kg and particle Q has mass 3m kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision P has speed 4u m s⁻¹ and Q has speed ku m s⁻¹, where k is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved. (a) Find the value of k. (b) Find, in terms of m and u, the magnitude of the impulse exerted on P by Q.

A-Level Edexcel Maths Mechanics Newtons Second Law: Particle P has mass m kg and par

Particle P has mass m kg and particle Q has mass 3m kg - Edexcel - A-Level Maths Mechanics - Question 2 - 2010 - Paper 1

Question 2

Particle P has mass m kg and particle Q has mass 3m kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. I... show full transcript

Worked Solution & Example Answer:Particle P has mass m kg and particle Q has mass 3m kg - Edexcel - A-Level Maths Mechanics - Question 2 - 2010 - Paper 1

Step 1

Find the value of k.

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Answer

To find the value of k, we can apply the principle of conservation of momentum. Before the collision, the total momentum of the system is:

$p_{initial} = 4u m - ku(3m)$

After the collision, the particles reverse direction and halve their speeds, so the total momentum becomes:

$p_{final} = -2(4u)m + (-3)(ku/2)$

Setting the initial momentum equal to the final momentum gives:

$4um - 3mku = -8um - (3/2)ku$

Simplifying and solving for k:

$4um - 3mku + 8um + (3/2)ku = 0$

Combine the terms:

$12um = 3mk + (3/2)ku$

This simplifies to:

$k = \frac{4}{3}$

Step 2

Find, in terms of m and u, the magnitude of the impulse exerted on P by Q.

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Answer

The impulse exerted on P by Q is equal to the change in momentum of particle P. To find this, we first calculate the initial and final momentum of P:

Initial momentum of P, $p_i = mu(4)$ (since it moves in the positive direction before the collision).

After the collision, the speed of P becomes $-2u$ , so the final momentum will be:

Final momentum of P, $p_f = m(-2u)$ .

The impulse $I$ is given by:

$I = p_f - p_i$

Substituting the values:

$I = m(-2u) - m(4u) = -6mu$

Taking the magnitude:

$|I| = 6mu$

Particle P has mass m kg and particle Q has mass 3m kg - Edexcel - A-Level Maths Mechanics - Question 2 - 2010 - Paper 1

Worked Solution & Example Answer:Particle P has mass m kg and particle Q has mass 3m kg - Edexcel - A-Level Maths Mechanics - Question 2 - 2010 - Paper 1

Find the value of k.

Find, in terms of m and u, the magnitude of the impulse exerted on P by Q.

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