M4 January 2006 Q5
5. Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(km\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of 60\(^\circ\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\tfrac{1}{2}\).
(a) Show that the speed of \(B\) immediately after the collision is \(\dfrac{3u}{4(k + 1)}\). (6)
Immediately after the collision the direction of motion of \(A\) makes an angle \(\arctan\left(2\sqrt{3}\right)\) with the direction of motion of \(B\).
(b) Show that \(k = \tfrac{1}{2}\). (6)
(c) Find the loss of kinetic energy due to the collision. (4)
| Scheme | Marks |
|---|---|
| \(CLM(\leftrightarrow)\colon\ mu\cos 60 = mv + kmw\) | M1 A1 |
| \(NLI\colon\ \dfrac{1}{2}u\cos 60 = w - v\) | M1 A1 |
| Solve for w: \(\ (1 + k)w = \dfrac{1}{2}u\left(1 + \dfrac{1}{2}\right)\) | M1 |
| \(\Rightarrow w = \dfrac{3u}{4(k + 1)}\qquad (*)\) | A1 |
| (6) |
| Scheme | Marks |
|---|---|
| Solve for v \(\ \rightarrow v = \dfrac{u(2 - k)}{4(k + 1)}\) | M1 A1 |
| \(\tan\theta = 2\sqrt{3} = \dfrac{u\sin 60}{v}\) | M1 A1 |
| \(= \dfrac{u\sqrt{3}}{2}.\dfrac{4(k + 1)}{u(2 - k)}\) | |
| Solve k: \(\ \rightarrow k = \dfrac{1}{2}\) | M1 A1 |
| (6) |
| Scheme | Marks |
|---|---|
| \(k = \dfrac{1}{2} \Rightarrow v = \dfrac{u}{4},\ w = \dfrac{u}{2}\) | B1 |
| KE loss \(= \dfrac{1}{2}mu^2 - \left(\dfrac{1}{2}m.\dfrac{u^2}{16} + \dfrac{1}{2}m.\dfrac{3u^2}{4} + \dfrac{1}{2}.\dfrac{1}{2}m.\dfrac{u^2}{4}\right)\) | M1 A1 |
| \(= \dfrac{1}{2}mu^2\left(1 - \dfrac{1}{16} - \dfrac{3}{4} - \dfrac{1}{8}\right)\) | |
| \(= \dfrac{1}{32}mu^2\) | A1 |
| (4) | |
| (16 marks) |