M2 January 2005 Q6
6. A particle \(P\) of mass \(3m\) is moving with speed \(2u\) in a straight line on a smooth horizontal table. The particle \(P\) collides with a particle \(Q\) of mass \(2m\) moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
(a) Show that the speed of \(Q\) after the collision is \(\tfrac{1}{5}u(9e + 4)\). (5)
As a result of the collision, the direction of motion of \(P\) is reversed.
(b) Find the range of possible values of \(e\). (5)
Given that the magnitude of the impulse of \(P\) on \(Q\) is \(\tfrac{32}{5}mu\),
(c) find the value of \(e\). (4)
| Scheme | Marks |
|---|---|
| LM \(6mu - 2mu = 3mx + 2my\) | M1 A1 |
| NEL \(y - x = 3eu\) | B1 |
| Solving to \(y = \tfrac{1}{5}u(9e + 4)\ \ *\) cso | M1 A1 |
| (5) |
| Scheme | Marks |
|---|---|
| Solving to \(x = \tfrac{2}{5}u(2 - 3e)\) oe | M1 A1 |
| \(x < 0\ \Rightarrow\ e > \tfrac{2}{3}\) | M1 A1 |
| \(\tfrac{2}{3} < e \leqslant 1\) ft their \(e\) for glb | A1ft |
| (5) |
| Scheme | Marks |
|---|---|
| \(2m\left[\tfrac{1}{5}u(9e + 4) + u\right] = \tfrac{32}{5}mu\) | M1 A1 |
| Solving to \(e = \tfrac{7}{9}\) awrt 0.78 | M1 A1 |
| (4) | |
| (14 marks) |