M2 January 2013 Q7
7. A particle \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal floor when it collides directly with another particle \(B\), of mass \(3m\), which is at rest on the floor. The coefficient of restitution between the particles is \(e\). The direction of motion of \(A\) is reversed by the collision.
After being struck by \(A\) the particle \(B\) collides directly with another particle \(C\), of mass \(4m\), which is at rest on the floor. The coefficient of restitution between \(B\) and \(C\) is \(2e\). Given that the direction of motion of \(B\) is reversed by this collision,
If the signs on their diagram and in their working are inconsistent, ignore the diagram. Penalise inconsistency between the two equations in the second accuracy mark.
| Scheme | Marks |
|---|---|
 | |
| \(mu = -mv + 3mw\) | M1 |
| \(u = -v + 3w\) | A1 |
| \(eu = w + v\) | M1 A1 |
| \(w = \dfrac{u}{4}(1 + e)\) | DM1 A1 |
| \(v = -w + eu = \dfrac{u}{4}(3e - 1)\) | A1 |
| (7) |
Notes
M1 CLM. Allow for \(v\) in either direction. Needs all 3 terms. Condone sign errors.
A1 \(v\) in either direction. Ignore diagram if equations "correct" but inconsistent with diagram.
M1 Impact law. Must be the right way round, but condone sign errors
A1 Correct equation. Signs consistent with CLM equn.
DM1 Solve for \(v\) or \(w\).
A1 One correct
A1 Both correct. \(1 - 3e \rightarrow\) A0 for \(v\)
If the signs on their diagram and in their working are inconsistent, ignore the diagram. Penalise inconsistency between the two equations in the B mark.
| Scheme | Marks |
|---|---|
 | |
| \(3mw = 4mX - 3mY\) | M1 A1ft |
| \(2ew = X + Y\) | B1ft |
| \(7Y = w(8e - 3)\) Or \(2ue(1 + e) - \dfrac{3u}{4}(1 + e) = 7Y\) | DM1 |
| \(\rightarrow e > \dfrac{3}{8}\) | A1 |
| \(Y > 0 \rightarrow \dfrac{3}{8} < e \leqslant \dfrac{1}{2}\) | A1 |
| (6) |
Notes
M1 CLM for \(w\).
A1ft Correct unsimplified (their \(w\))
B1ft Impact law. Must be the right way up. Their \(w\)
DM1 Solve for (7)\(Y\)
A1 NB No longer ft. Condone \(<\).
(Corrected from the printed mark scheme: \(7Y = w(8e - 3)\) is printed with a capital \(W\).)
| Scheme | Marks |
|---|---|
| \(\dfrac{u}{28}(1 + e)(8e - 3) > \dfrac{u}{4}(3e - 1)\) \(2e^2 - 4e + 1 > 0\) | M1 |
| \(e = \dfrac{4 \pm \sqrt{16 - 8}}{4} = 1.707,\ 0.293\) | DM1 |
| \(2e^2 - 4e + 1 < 0\) for \(\dfrac{3}{8} < e \leqslant \dfrac{1}{2}\) so no second collision. | A1 |
| (3) | |
| (16 marks) |
Notes
M1 For a second collision their \(Y >\) their \(v\)
DM1 Obtain the critical values
A1 Compare 0.293 (o.e.) with \(\dfrac{3}{8}\) to reach correct conclusion for correct reason.