M4 June 2014 (R) Q1
1. A small smooth ball of mass \(m\) is falling vertically when it strikes a fixed smooth plane which is inclined to the horizontal at an angle \(\alpha\), where \(0^\circ < \alpha < 45^\circ\). Immediately before striking the plane the ball has speed \(u\). Immediately after striking the plane the ball moves in a direction which makes an angle of 45\(^\circ\) with the plane. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(m\), \(u\) and \(e\), the magnitude of the impulse of the plane on the ball. (11)
| Scheme | Marks |
|---|---|
| \(v\cos 45^\circ = u\sin\alpha\) parallel | M1 A1 |
| \(v\sin 45^\circ = eu\cos\alpha\) perpendicular | M1 A1 |
| \(e = \tan\alpha\) or square & add | M1 A1 |
| \(I = m(v\cos 45^\circ + u\cos\alpha)\) impulse | M1 A1 |
| \(= mu(\sin\alpha + \cos\alpha)\) in terms of \(u,\ \alpha\) | M1 |
| \(= \dfrac{mu(1 + e)}{\sqrt{1 + e^2}}\) in terms of \(u,\ e\) | M1 A1 |
| (11 marks) |