M2 January 2006 Q4
4. A particle \(A\) of mass \(2m\) is moving with speed \(3u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2u\) in the opposite direction to \(A\). Immediately after the collision the speed of \(B\) is \(\tfrac{8}{3}u\) and the direction of motion of \(B\) is reversed.
(a) Calculate the coefficient of restitution between \(A\) and \(B\). (6)
(b) Show that the kinetic energy lost in the collision is \(7mu^2\). (3)
After the collision \(B\) strikes a fixed vertical wall that is perpendicular to the direction of motion of \(B\). The magnitude of the impulse of the wall on \(B\) is \(\tfrac{14}{3}mu\).
(c) Calculate the coefficient of restitution between \(B\) and the wall. (4)
| Scheme | Marks |
|---|---|
| LM \(6mu - 2mu = 2mx + \dfrac{8}{3}mu\) | M1 A1 |
| \(\left(x = \dfrac{2}{3}u\right)\) | |
| NEL \(\dfrac{8}{3}u - x = 5ue\) | M1 A1 |
| Solving to \(e = \dfrac{2}{5}\) | M1 A1 |
| (6) |
| Scheme | Marks |
|---|---|
| Initial K.E. \(= \dfrac{1}{2} \times 2m(3u)^2 + \dfrac{1}{2} \times m(2u)^2 = 11mu^2\) | |
| Final K.E. \(= \dfrac{1}{2} \times 2m\left(\dfrac{2}{3}u\right)^2 + \dfrac{1}{2} \times m\left(\dfrac{8}{3}u\right)^2 = 4mu^2\) both | M1 |
| Change in K.E. \(= 7mu^2\ \ *\) | M1 A1 |
| (3) |
Notes
M1 Subtracting and simplifying to \(kmu^2\) A1cso
| Scheme | Marks |
|---|---|
| \(m\left(\dfrac{8}{3}u + v\right) = \dfrac{14}{3}mu\) | M1 A1 |
| \((v = 2u)\) | |
| \(e = \dfrac{2}{\frac{8}{3}} = \dfrac{3}{4}\) | M1 A1 |
| (4) | |
| (13 marks) |