M2 June 2009 Q8
8. Particles \(A\), \(B\) and \(C\) of masses \(4m\), \(3m\) and \(m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected towards each other with speeds \(u\) m s\(^{-1}\) and \(v\) m s\(^{-1}\) respectively, and collide directly.
As a result of the collision, \(A\) is brought to rest and \(B\) rebounds with speed \(kv\) m s\(^{-1}\). The coefficient of restitution between \(A\) and \(B\) is \(\dfrac{3}{4}\).
(a) Show that \(u = 3v\). (6)
(b) Find the value of \(k\). (2)
Immediately after the collision between \(A\) and \(B\), particle \(C\) is projected with speed \(2v\) m s\(^{-1}\) towards \(B\) so that \(B\) and \(C\) collide directly.
(c) Show that there is no further collision between \(A\) and \(B\). (4)
| Scheme | Marks |
|---|---|
| Conservation of momentum: \(\ 4mu - 3mv = 3mkv\) | M1A1 |
| Impact law: \(\ kv = \dfrac{3}{4}(u + v)\) | M1A1 |
| Eliminate k: \(\ 4mu - 3mv = 3m \times \dfrac{3}{4}(u + v)\) | DM1 |
| \(u = 3v\) (Answer given) | A1 |
| (6) |
| Scheme | Marks |
|---|---|
| \(kv = \dfrac{3}{4}(3v + v)\), \(k = 3\) | M1,A1 |
| (2) |
| Scheme | Marks |
|---|---|
| Impact law: \(\ (kv + 2v)e = v_C - v_B\ \ \ (5ve = v_C - v_B)\) | B1 |
| Conservation of momentum : \(\ 3 \times kv - 1 \times 2v = 3v_B + v_c\ \ \ (7v = 3v_B + v_c)\) | B1 |
| Eliminate \(v_C\) : \(\ v_B = \dfrac{v}{4}(7 - 5e) > 0\) hence no further collision with \(A\). | M1 A1 |
| (4) | |
| (12 marks) |