M4 June 2018 Q7
7. Two smooth uniform spheres \(A\) and \(B\), of mass 2 kg and 3 kg respectively, and of equal radius, are moving on a smooth horizontal plane when they collide.
Immediately before the collision the velocity of \(A\) is \((3\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) and the velocity of \(B\) is \((-\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). Immediately after the collision the velocity of \(A\) is \((\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\).
(a) Show that, at the instant when \(A\) and \(B\) collide, their line of centres is parallel to \(-\mathbf{i} + \mathbf{j}\). (4)
(b) Find the velocity of \(B\) immediately after the collision. (3)
(c) Find the coefficient of restitution between \(A\) and \(B\). (6)
| Scheme | Marks |
|---|---|
| Impulse on \(A\): \(I = 2(\mathbf{i} + 3\mathbf{j} - 3\mathbf{i} - \mathbf{j})\) | M1A1 |
| \(= -4\mathbf{i} + 4\mathbf{j} = 4(-\mathbf{i} + \mathbf{j})\) | A1 |
| Impulse parallel to l.o.c., hence l.o.c. parallel to \(-\mathbf{i} + \mathbf{j}\) (Given answer) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| Impulse equal and opposite: \(4\mathbf{i} - 4\mathbf{j} = 3(\mathbf{v} + \mathbf{i} - 2\mathbf{j})\) | M1A1 |
| \(3\mathbf{v} = \mathbf{i} + 2\mathbf{j},\quad \mathbf{v} = \dfrac{1}{3}(\mathbf{i} + 2\mathbf{j})\) | A1 |
| (3) |
Alt using CLM
| \(2(3\mathbf{i} + \mathbf{j}) + 3(-\mathbf{i} + 2\mathbf{j}) = 2(\mathbf{i} + 3\mathbf{j}) + 3\mathbf{v}\) | M1A1 |
| \(3\mathbf{v} = \mathbf{i} + 2\mathbf{j},\quad \mathbf{v} = \dfrac{1}{3}(\mathbf{i} + 2\mathbf{j})\) | A1 |
| Scheme | Marks |
|---|---|
| Components of velocities parallel to \(-\mathbf{i} + \mathbf{j}\): \(A\) before : \((3\mathbf{i} + \mathbf{j})\cdot\dfrac{1}{\sqrt{2}}(-\mathbf{i} + \mathbf{j}) = \dfrac{-2}{\sqrt{2}}\) \(A\) after : \((\mathbf{i} + 3\mathbf{j})\cdot\dfrac{1}{\sqrt{2}}(-\mathbf{i} + \mathbf{j}) = \dfrac{2}{\sqrt{2}}\) \(B\) before : \((-\mathbf{i} + 2\mathbf{j})\cdot\dfrac{1}{\sqrt{2}}(-\mathbf{i} + \mathbf{j}) = \dfrac{3}{\sqrt{2}}\) \(B\) after : \(\dfrac{1}{3}(\mathbf{i} + 2\mathbf{j})\cdot\dfrac{1}{\sqrt{2}}(-\mathbf{i} + \mathbf{j}) = \dfrac{1}{3\sqrt{2}}\) follow through from 7(b) NB: the marks are all available if the unit vector \(\left(\sqrt{2}\right)\) is not used. | M1A3 |
| Coefficient of restitution: \(\dfrac{1}{\sqrt{2}}\left(2 - \dfrac{1}{3}\right) = \dfrac{e}{\sqrt{2}}(3 + 2)\) | M1 |
| \(e = \dfrac{1}{3}\) | A1 |
| (6) | |
| (13 marks) |
Alternative (non-vector form)
| Components parallel to the line of centres: | M1A3 |
| \(A\) before: \(-\sqrt{10}\cos(135 - \beta) = -\sqrt{10}\left(-\dfrac{1}{\sqrt{2}}\cdot\dfrac{1}{\sqrt{10}} + \dfrac{1}{\sqrt{2}}\cdot\dfrac{3}{\sqrt{10}}\right) = -\dfrac{2}{\sqrt{2}}\) | |
| \(A\) after: \(\sqrt{10}\cos(135 - \beta) = \sqrt{10}\left(-\dfrac{1}{\sqrt{2}}\cdot\dfrac{1}{\sqrt{10}} + \dfrac{1}{\sqrt{2}}\cdot\dfrac{3}{\sqrt{10}}\right) = \dfrac{2}{\sqrt{2}}\) | |
| \(B\) before: \(\sqrt{5}\cos(45 - \alpha) = \sqrt{5}\left(\dfrac{1}{\sqrt{2}}\cdot\dfrac{2}{\sqrt{5}} + \dfrac{1}{\sqrt{2}}\cdot\dfrac{1}{\sqrt{5}}\right) = \dfrac{3}{\sqrt{2}}\) | |
| \(B\) after: \(\dfrac{\sqrt{5}}{3}\cos(45 + \alpha) = \dfrac{\sqrt{5}}{3}\left(\dfrac{1}{\sqrt{2}}\cdot\dfrac{2}{\sqrt{5}} - \dfrac{1}{\sqrt{2}}\cdot\dfrac{1}{\sqrt{5}}\right) = \dfrac{1}{3\sqrt{2}}\) follow through from 7(b) | |
| Coefficient of restitution: \(\dfrac{1}{\sqrt{2}}\left(2 - \dfrac{1}{3}\right) = \dfrac{e}{\sqrt{2}}(3 + 2)\) | M1 |
| \(e = \dfrac{1}{3}\) | A1 |