M2 June 2011 Q2

EdexcelOld spec8 marksCollisions: 2 Spheres Direct

2. A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal surface with speed \(4u\). The particle \(P\) collides directly with a particle \(Q\) of mass \(3m\) which is at rest on the surface. The coefficient of restitution between \(P\) and \(Q\) is \(e\). The direction of motion of \(P\) is reversed by the collision.

Show that \(e > \dfrac{1}{3}\). (8)

Diagram: P (m) moving at 4u towards Q (3m) at rest; after the collision P moves at v to the left and Q at x to the right — Mark scheme

Scheme	Marks
\(4mu = 3mx - mv\)	M1 A1
\(4ue = x + v\)	M1 A1
\(4u = 3(4ue - v) - v\)
\(4u = 12ue - 4v\)
\(v = (3e - 1)u\)	DM1 A1
\(v > 0 \Rightarrow 3e > 1\)	DM1
\(\therefore e > \frac{1}{3}\quad **\)	A1
	(8 marks)