M4 June 2013 (R) Q2

EdexcelOld spec6 marksCollisions: 1 Sphere Oblique

Figure 1: ball B falling vertically with speed u m/s onto a plane inclined at angle alpha to the horizontal — Figure 1

A smooth fixed plane is inclined at an angle \(\alpha\) to the horizontal. A smooth ball \(B\) falls vertically and hits the plane. Immediately before the impact the speed of \(B\) is \(u\) m s\(^{-1}\), as shown in Figure 1. Immediately after the impact the direction of motion of \(B\) is horizontal. The coefficient of restitution between \(B\) and the plane is \(\dfrac{1}{3}\).

Find the size of angle \(\alpha\). (6)

Ball arriving vertically with speed u and leaving horizontally with speed v from plane at angle alpha — Mark scheme

Scheme	Marks
CLM: \(\quad u\sin\alpha = v\cos\alpha\)	M1 A1
Impact: \(\quad \dfrac{1}{3}u\cos\alpha = v\sin\alpha\)	M1 A1
\(\dfrac{1}{3} \times \dfrac{1}{\tan\alpha} = \tan\alpha\)	M1
\(\tan\alpha = \dfrac{1}{\sqrt{3}}\)
\(\alpha = 30^\circ\ \left(\text{or } \frac{\pi}{6} \text{ or } 0.52\,\text{rad}\right)\)	A1
	(6)
	(6 marks)

Notes