M2 June 2015 Q8
8. Three identical particles \(P\), \(Q\) and \(R\), each of mass \(m\), lie in a straight line on a smooth horizontal plane with \(Q\) between \(P\) and \(R\). Particles \(P\) and \(Q\) are projected directly towards each other with speeds \(4u\) and \(2u\) respectively, and at the same time particle \(R\) is projected along the line away from \(Q\) with speed \(3u\). The coefficient of restitution between each pair of particles is \(e\). After the collision between \(P\) and \(Q\) there is a collision between \(Q\) and \(R\).
It is given that \(e = \dfrac{3}{4}\)
| Scheme | Marks |
|---|---|
| \(4mu - 2mu = mv + mw\) | M1 A1 |
| \(w - v = 6eu\) | M1 A1 |
| \(v + w = 2u\) \(w - v = 6eu\ \ \ \ \ 2w = 2u + 6eu,\ \ (w = u + 3eu)\) | DM1 |
| For \(Q\) and \(R\) to collide require \(w > 3u\), | M1 |
| \(u + 3eu > 3u,\ \ \ \ \ e > \dfrac{2}{3}\) | A1 |
| (7) |
Notes
M1 Equation for CLM. Requires all 4 terms. Condone sign errors. Condone \(m\) missing throughout.
M1 Impact law. \(e\) must be used correctly. Condone sign errors
A1 Signs should be consistent with equation for CLM.
DM1 Solve for \(w\). Dependent on the two previous M marks.
M1 Use inequality to compare their \(w\) with \(3u\).
A1 Reach *Given answer* with no errors seen.
8a alt
| Collision between \(Q\) and \(R \Rightarrow w > 3u\) | M1 |
| Magnitude of impulse on \(Q > 5mu\) Magnitude of impulse on \(P > 5mu\) | A1 |
| \(\Rightarrow v < -u\) | M1 |
| \(e = \dfrac{w - v}{4u + 2u}\) | M1 |
| Speed of separation after collision \(> u + 3u\) | M1 |
| \(e > \dfrac{4u}{4u + 2u}\) | A1 |
| \(e > \dfrac{2}{3}\) | A1 |
M1 Impact law
A1 Reach given inequality with no errors seen.
| Scheme | Marks |
|---|---|
| \(w = u + 3eu = \dfrac{13}{4}u,\ \ v = -\dfrac{5}{4}u\) | B1 |
| \(mw + 3mu = mx + my\) \(\dfrac{3}{4}(w - 3u) = y - x\) | M1 A1 |
| \(\dfrac{25}{4}u = x + y,\ \ \dfrac{3u}{16} = y - x\) | |
| Solve for \(x\) (or \(kx\)) | DM1 |
| \(\dfrac{97}{16}u = 2x,\ \ x = \dfrac{97u}{32}\ (= 3.03125u)\) | A1 |
| \(P\) and \(Q\) moving away from each other, so no collision. | A1 |
| (6) | |
| (13 marks) |
Notes
B1 Correct values seen or implied.
M1 Equations for CLM and impact. Allow with \(w\) or their \(w\). All terms required. Condone sign errors. \(e\) must be used correctly.
A1 Correct equations in \(w\) or their \(w\).
DM1 Dependent on the previous M1. Need to get far enough to give a convincing concluding argument.
A1 Correct expression for \(kx\)
A1 Reach given conclusion with no errors seen.