M4 June 2012 Q1
1. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan\alpha = \dfrac{3}{4}\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\dfrac{3}{4}\).
| Scheme | Marks |
|---|---|
 | |
| \(mu\cos\alpha = mw + 2mV\) | M1 A1 |
| \(eu\cos\alpha = -w + V\) | M1 A1 |
| \(u\cos\alpha(e + 1) = 3V \Rightarrow\) (i) \(u = \dfrac{15V}{7}\) | M1 A1 |
| \(\Rightarrow w = -\dfrac{2V}{7}\) | A1 |
| (ii) speed of \(S = \sqrt{\left(\frac{-2V}{7}\right)^2 + (u\sin\alpha)^2} = \dfrac{V\sqrt{85}}{7}\) | M1 A1 |
| (9) |
Notes
M1 CLM parallel to the line of centres. \(\dfrac{4}{5}u = w + 2V\). Need all terms but condone sign errors.
M1 Impact law. Must be the right way round. \(\dfrac{3}{4} \times \dfrac{4}{5}u = V - w\)
M1 Eliminate \(w\) and solve for \(u\) in terms of \(V\) or v.v.
A1 \(2.14V\) or better
A1 Solve for \(w\) in terms of \(V\). \(-0.286V\) or better
M1 Use of Pythagoras with their \(u\sin\alpha\) and \(w\). \(\sqrt{\left(\dfrac{-2V}{7}\right)^2 + \left(\dfrac{15V}{7} \times \dfrac{3}{5}\right)^2}\)
A1 \(\sqrt{\dfrac{85}{49}}V\), accept \(1.32V\) or better
| Scheme | Marks |
|---|---|
| \(\tan\theta = \dfrac{\frac{9V}{7}}{\frac{2V}{7}} = \dfrac{9}{2}\) | M1 A1 |
| defln angle \(= 180^\circ - (\theta + \alpha)\) | DM1 |
| \(= 65.7^\circ\) (3 sf) | A1 |
| (4) | |
| (13 marks) |
Notes
M1 Direction of \(S\) after the collision. Condone \(\dfrac{2}{9}\)
A1 \(77.5^\circ\) or \(12.5^\circ\) seen or implied
DM1 Combine their \(\theta\) and \(\alpha\) to find the required angle. e.g. \(12.5^\circ + \tan^{-1}\left(\dfrac{4}{3}\right)\)
A1 Accept \(66^\circ\)