M4 June 2009 Q5
5. Two small smooth spheres \(A\) and \(B\), of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \((\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and the velocity of \(B\) is \(-2\mathbf{i}\) m s\(^{-1}\). Immediately after the collision the velocity of \(A\) is \(\mathbf{j}\) m s\(^{-1}\).
(a) Show that the velocity of \(B\) immediately after the collision is \(2\mathbf{j}\) m s\(^{-1}\). (3)
(b) Find the impulse of \(B\) on \(A\) in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to \(\mathbf{i} + \mathbf{j}\). (4)
(c) Find the coefficient of restitution between \(A\) and \(B\). (6)
| Scheme | Marks |
|---|---|
| CLM: \(\ 2(\mathbf{i} + 2\mathbf{j}) + -2\mathbf{i} = 2\mathbf{j} + \mathbf{v}\) | M1 A1 |
| \(\mathbf{v} = 2\mathbf{j}\) m s\(^{-1}\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathbf{I} = 2(\mathbf{j} - (\mathbf{i} + 2\mathbf{j}))\) | M1 A1 |
| \(= (-2\mathbf{i} - 2\mathbf{j})\) Ns | A1 |
| Since \(\mathbf{I}\) acts along l.o.c.c., l.o.c.c is parallel to \(\mathbf{i} + \mathbf{j}\) | B1 |
| (4) |
| Scheme | Marks |
|---|---|
| Before \(A\): \(\ (\mathbf{i} + 2\mathbf{j})\cdot\dfrac{1}{\sqrt{2}}(\mathbf{i} + \mathbf{j}) = \dfrac{3}{\sqrt{2}}\) \(B\): \(\ -2\mathbf{i}\cdot\dfrac{1}{\sqrt{2}}(\mathbf{i} + \mathbf{j}) = \dfrac{-2}{\sqrt{2}}\) After \(A\): \(\ \mathbf{j}\cdot\dfrac{1}{\sqrt{2}}(\mathbf{i} + \mathbf{j}) = \dfrac{1}{\sqrt{2}}\) \(B\): \(\ 2\mathbf{j}\cdot\dfrac{1}{\sqrt{2}}(\mathbf{i} + \mathbf{j}) = \dfrac{2}{\sqrt{2}}\) | M1 A3 |
| NIL: \(\ e = \dfrac{\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}}{\frac{3}{\sqrt{2}} - \frac{-2}{\sqrt{2}}} = \dfrac{1}{5}\) | DM1 A1 |
| (6) | |
| (13 marks) |
Notes
(Corrected from the printed mark scheme: the component of \(B\)’s velocity before the collision is printed as \(-2\mathbf{j}\cdot\dfrac{1}{\sqrt{2}}(\mathbf{i} + \mathbf{j})\); the velocity of \(B\) is \(-2\mathbf{i}\).)