M2 June 2008 Q2
2. A particle \(A\) of mass \(4m\) is moving with speed \(3u\) in a straight line on a smooth horizontal table. The particle \(A\) collides directly with a particle \(B\) of mass \(3m\) moving with speed \(2u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Immediately after the collision the speed of \(B\) is \(4eu\).
(a) Show that \(e = \dfrac{3}{4}\). (5)
(b) Find the total kinetic energy lost in the collision. (4)
| Scheme | Marks |
|---|---|
| LM \(12mu + 6mu = 4mx + 12meu\) | B1 |
| NEL \(4eu - x = eu\) | M1 A1 |
| Eliminating \(x\) to obtain equation in \(e\) | DM1 |
| Leading to \(e = \dfrac{3}{4}\) * cso | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(x = 3eu\) or \(\dfrac{9}{4}u\) or \(4.5u - 3eu\) seen or implied in (b) | B1 |
| Loss in KE \(= \dfrac{1}{2}4m(3u)^2 + \dfrac{1}{2}3m(2u)^2 - \dfrac{1}{2}4m\left(\dfrac{9}{4}u\right)^2 - \dfrac{1}{2}3m(3u)^2\) ft their \(x\) | M1 A1ft |
| \(= 24mu^2 - 23\tfrac{5}{8}mu^2 = \dfrac{3}{8}mu^2 = 0.375mu^2\) | A1 |
| (4) | |
| (9 marks) |