M4 June 2005 Q3

EdexcelOld spec11 marksCollisions: 2 Spheres Oblique

Figure 1: sphere Q touching sphere P along the line of centres; Q approaches at angle alpha to the line of centres and leaves at angle beta to it — Figure 1

A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

Show that \((1 - e)\tan\beta = 2\tan\alpha\). (11)

Spheres Q and P, each of mass m: before, Q has components u cos alpha along and u sin alpha perpendicular to the line of centres, P at rest; after, Q has v1 along and v3 perpendicular, at angle beta, P has v2 along and 0 perpendicular — Mark scheme

Scheme	Marks
\(v_3 = u\sin\alpha\)	B1
CLM: \(\quad v_1 + v_2 = u\cos\alpha\)	M1 A1
NIL: \(\quad -v_1 + v_2 = eu\cos\alpha\)	M1 A1
\(\dfrac{v_3}{v_1} = \tan\beta\)	M1 A1
elim \(v_2\)	M1
elim \(v_3\)	M1
elim \(u\)	M1
\(\Rightarrow \tan\beta(1 - e) = 2\tan\alpha\ \ *\)	A1
	(11)

Notes