AS June 2019 Q4

EdexcelAS paperCurrent spec10 marksCollisions: 2 Spheres Direct

4. 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 \(e\).

(a) Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. (9)

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

(b) write down, in terms of \(u\), \(k\) and \(e\), the speed of \(Q\) after this second collision. (1)

Diagram: P (km), Q (m) and R (km); Q moving towards R with speed u; after the collision Q moves back towards P with speed v_Q and R moves away with speed v_R — Mark scheme (a)

Scheme	Marks	AO

Use of conservation of momentum	M1	3.1a
\(mu = -mv_Q + kmv_R\)	A1	1.1b
Use of NLR	M1	3.4
\(eu = v_Q + v_R\)	A1	1.1b
Using correct strategy to solve problem by finding \(v_Q\)	M1	3.1a
\(v_Q = \dfrac{u(ke-1)}{k+1}\) or \(v_Q = \dfrac{v_R(ke-1)}{1+e}\)	A1	1.1b
For second collision, \(v_Q \gt 0\)	M1	3.1a
\(\dfrac{u(ke-1)}{k+1} \gt 0\)	M1	1.1b
\(k \gt \dfrac{1}{e}\)	A1	1.1b
	(9)

Scheme	Marks	AO
\(\dfrac{u(ke-1)^2}{(k+1)^2}\)	B1	2.2a
	(1)
	(10 marks)

AS June 2019 Q4

AS June 2019 Q4

Notes

Notes