M2 June 2012 Q2
2. A particle \(P\) of mass \(3m\) is moving with speed \(2u\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass \(4m\) moving on the plane with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
Given that the direction of motion of \(P\) is reversed by the collision,
| Scheme | Marks |
|---|---|
| \(3m.2u - 4mu = 3mv_1 + 4mv_2\) | M1 A1 |
| \(e(2u + u) = -v_1 + v_2\) | M1 A1 |
| \(\dfrac{u(2 + 9e)}{7} = v_2\) | DM1 A1 |
| (6) |
Notes
M1 CLM. Need all terms. Condone sign slips.
A1 Correct but check their directions for \(v_1\) & \(v_2\).
M1 Impact law. Must be used the right way round, but condone sign slips.
A1 Directions of \(v_1\) & \(v_2\) must be consistent between the two equations. (Ignore the diagram if necessary)
DM1 Eliminate \(v_1\) to produce an equation in \(v_2\) only. Dependent on both previous M marks – must be using both equations.
A1 DO NOT accept the negative. The question asks for speed.
| Scheme | Marks |
|---|---|
| \(v_1 = \dfrac{2u(1 - 6e)}{7}\) | M1 A1 |
| \(v_1 < 0 \Rightarrow e > \dfrac{1}{6}\) | DM1 A1 |
| \(1 \geqslant e > \dfrac{1}{6}\) | B1 |
| (5) | |
| (11 marks) |
Notes
M1 Use the work from (a) or restart to find \(v_1\) or \(\lambda v_1\) for a constant \(\lambda\). If using work from (a) this mark is dependent on the first 2 M marks.
A1 a.e.f. Correct for their direction. Allow for \(\lambda v_1\)
DM1 An appropriate inequality for their \(v_1\) (seen or implied) – requires previous M1 scored. Work on \(v_1 = 0\) scores M0 until the inequality is formed.
A1 Accept \(\dfrac{2}{12}\). Answer must follow from correct work for \(v_1\)
B1 For (their value) \(< e \leqslant 1\)
SR: from \(v_1 \leqslant 0\) could score M1A0B1