M4 June 2008 Q4

EdexcelOld spec8 marksCollisions: 1 Sphere Oblique

Figure 1: ball approaching a vertical wall at angle 2 theta to the wall and rebounding at angle theta to the wall — Figure 1

A small smooth ball \(B\), moving on a horizontal plane, collides with a fixed vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(2\theta\), where \(0^\circ \lt \theta \lt 45^\circ\). Immediately after the collision the angle between the direction of motion of \(B\) and the wall is \(\theta\), as shown in Figure 1.

Given that the coefficient of restitution between \(B\) and the wall is \(\frac{3}{8}\), find the value of \(\tan\theta\). (8)

Mark scheme

Scheme	Marks
\(u\cos 2\theta = v\cos\theta\)	M1 A1
\(\tfrac{3}{8}u\sin 2\theta = v\sin\theta\)	M1 A1
\(3\tan 2\theta = 8\tan\theta\)	M1
\(\dfrac{6\tan\theta}{1 - \tan^2\theta} = 8\tan\theta\)	M1
\(\tan^2\theta = \tfrac{1}{4}\quad (\tan\theta \neq 0)\)
\(\tan\theta = \tfrac{1}{2}\)	M1 A1
	(8 marks)