M2 June 2007 Q7
7. Two small spheres \(P\) and \(Q\) of equal radius have masses \(m\) and \(5m\) respectively. They lie on a smooth horizontal table. Sphere \(P\) is moving with speed \(u\) when it collides directly with sphere \(Q\) which is at rest. The coefficient of restitution between the spheres is \(e\), where \(e > \dfrac{1}{5}\).
Three small spheres \(A\), \(B\) and \(C\) of equal radius lie at rest in a straight line on a smooth horizontal table, with \(B\) between \(A\) and \(C\). The spheres \(A\) and \(C\) each have mass \(5m\), and the mass of \(B\) is \(m\). Sphere \(B\) is projected towards \(C\) with speed \(u\). The coefficient of restitution between each pair of spheres is \(\dfrac{4}{5}\).
| Scheme | Marks |
|---|---|
| CLM: \(mv + 5mw = mu\) | B1 |
| NLI: \(w - v = eu\) | B1 |
| Solve \(v\): \(\ v = \tfrac{1}{6}(1 - 5e)u\), so speed \(= \dfrac{1}{6}(5e - 1)u\) (NB – answer given on paper) | M1* A1 |
| Solve \(w\): \(\ w = \tfrac{1}{6}(1 + e)u\) | M1* A1 |
| * The M’s are dependent on having equations (not necessarily correct) for CLM and NLI | |
| (6) |
Notes
B1 Conservation of momentum – signs consistent with their diagram/between the two equations
B1 Impact equation
M1 Attempt to eliminate w
A1 correct expression for v. Q asks for speed so final answer must be verified positive with reference to \(e > 1/5\).
Answer given so watch out for fudges.
M1 Attempt to eliminate v
A1 correct expression for w
| Scheme | Marks |
|---|---|
| After \(B\) hits \(C\), velocity of \(B\) = “\(v\)” \(= \tfrac{1}{6}\left(1 - 5 \cdot \tfrac{4}{5}\right)u = -\tfrac{1}{2}u\) | M1 A1 |
| velocity \(< 0 \Rightarrow\) change of direction \(\Rightarrow B\) hits \(A\) | A1 CSO |
| (3) |
Notes
M1 Substitute for e in speed or velocity of P to obtain \(\boldsymbol{v}\) in terms of \(\boldsymbol{u}\). Alternatively, can obtain v in terms of w
A1 (+/-) u/2 \(\left(v = -\dfrac{5w}{3}\right)\)
A1 CSO Justify direction (and correct conclusion)
| Scheme | Marks |
|---|---|
| velocity of \(C\) after \(= \tfrac{3}{10}u\) | B1 |
| When \(B\) hits \(A\), “\(u\)” \(= \tfrac{1}{2}u\), so velocity of \(B\) after \(= -\tfrac{1}{2}\left(-\tfrac{1}{2}u\right) = \tfrac{1}{4}u\) | B1 |
| Travelling in the same direction but \(\tfrac{1}{4} < \tfrac{3}{10} \Rightarrow\) no second collision | M1 A1 CSO |
| (4) | |
| (13 marks) |
Notes
B1 speed of C = value of w \(= (\pm)\dfrac{3u}{10}\) (Must be referred to in (c) to score the B1.)
B1 speed of B after second collision \((\pm)\dfrac{1}{4}u\) or \((\pm)\dfrac{5}{6}w\)
M1 Comparing their speed of B after 2nd collision with their speed of C after first collision.
A1 CSO. Correct conclusion.