M4 June 2006 Q2

EdexcelOld spec6 marksCollisions: 1 Sphere Oblique

2. A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a fixed smooth vertical wall. Immediately before the collision, the speed of \(S\) is \(U\) and its direction of motion makes an angle \(\alpha\) with the wall. The coefficient of restitution between \(S\) and the wall is \(e\). Find the kinetic energy of \(S\) immediately after the collision. (6)

Plan view: sphere approaches the wall with speed U at angle alpha and leaves with speed v at angle theta — Mark scheme

Scheme	Marks
\((\rightarrow)\quad U\cos\alpha = v\cos\theta\)	M1 A1
\((\uparrow)\quad eU\sin\alpha = v\sin\theta\)	M1 A1
\(\Rightarrow v^2 = U^2\left(\cos^2\alpha + e^2\sin^2\alpha\right)\)	M1
\(\Rightarrow KE = \dfrac{1}{2}mU^2\left(\cos^2\alpha + e^2\sin^2\alpha\right)\)	A1
	(6)

Notes