M2 June 2013 Q7
7. Three particles \(P\), \(Q\) and \(R\) lie at rest in a straight line on a smooth horizontal table with \(Q\) between \(P\) and \(R\). The particles \(P\), \(Q\) and \(R\) have masses \(2m\), \(3m\) and \(4m\) respectively. Particle \(P\) is projected towards \(Q\) with speed \(u\) and collides directly with it. The coefficient of restitution between each pair of particles is \(e\).
After the collision between \(P\) and \(Q\) there is a direct collision between \(Q\) and \(R\).
Given that \(e = \dfrac{3}{4}\), find
Immediately after the collision between \(Q\) and \(R\), the rate of increase of the distance between \(P\) and \(R\) is \(V\).
| Scheme | Marks |
|---|---|
| CLM: \(2mu = 2mv + 3mw\) | M1 A1 |
| Impact: \(w - v = eu\) | M1 A1 |
| Subst \(v = w - eu:\ 2u = 2(w - eu) + 3w = 5w - 2eu\) | DM1 |
| \(w = \dfrac{2}{5}(1 + e)u\ \ \ \)*Answer Given* | A1 |
| (6) |
Notes
M1 All three terms required, but condone sign errors
M1 Condone sign error, but must be subtracting and \(e\) must be used correctly.
A1 Penalise inconsistent signs here.
DM1 Solve for \(w\). Requires the two preceding M marks
(The brackets in \(2(w - eu)\) and \(\dfrac{2}{5}(1 + e)u\) are missing from the printed mark scheme.)
| Scheme | Marks |
|---|---|
| \(w = \dfrac{7u}{10}\) | B1 |
| CLM: \(3mw = 3mx + 4my\) and Impact: \(y - x = \dfrac{3w}{4}\) | M1A1 |
| Subst: \(3w = 3x + 4\left(x + \dfrac{3}{4}w\right)\) | DM1 |
| \(x = 0\), | A1 |
| \(y = \dfrac{3}{4}w = \dfrac{21}{40}u\) | A1 |
| (6) |
Notes
B1 Seen, or implied by correct speeds.
M1A1 Both needed
DM1 Solve for \(x\) or \(y\). Dependent on the preceding M mark
A1 \(0.525u\),
| Scheme | Marks |
|---|---|
| \(v = -\dfrac{u}{20}\) | B1 |
| Speed of separation \(= \dfrac{u}{20} + \dfrac{21u}{40} = \dfrac{23u}{40}\) | M1 A1 |
| (3) | |
| (15 marks) |
Notes
B1 Correct velocity of \(P\)
M1 Correct use of their values and substitute for \(e\). Check directions carefully
A1 \(0.575u\)