M4 January 2005 Q1
1. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors.]
Two smooth uniform spheres \(A\) and \(B\) have equal radius but masses \(m\) and \(5m\) respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocities of \(A\) and \(B\) are \((\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and \((-\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\) respectively. Immediately after the collision, the velocity of \(A\) is \((-2\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\).
(a) By considering the impulse on \(A\), find a unit vector parallel to the line joining the centres of the spheres when they collide. (4)
(b) Find the velocity of \(B\) immediately after the collision. (3)
| Scheme |
|---|
| Impulse on \(A\) is in the direction of the line of centres. |
| Impulse on \(A = \Delta(mv) = m(-2\mathbf{i} + 5\mathbf{j}) - m(\mathbf{i} + 2\mathbf{j}) = m(-3\mathbf{i} + 3\mathbf{j})\). |
| Therefore direction of line of centres is \((-\mathbf{i} + \mathbf{j})\). A unit vector in this direction is \(\dfrac{(-\mathbf{i} + \mathbf{j})}{\sqrt{2}}\). |
Notes
The published mark scheme for this paper is a set of worked answers: no mark allocation is printed.
| Scheme |
|---|
| Let velocity of \(b\) after collision be \(v_1\mathbf{i} + v_2\mathbf{j}\) |
| Momentum conserved: \(m(\mathbf{i} + 2\mathbf{j}) + 5m(-\mathbf{i} + 3\mathbf{j}) = m(-2\mathbf{i} + 5\mathbf{j}) + 5m(v_1\mathbf{i} + v_2\mathbf{j})\) |
| Cancel \(m\) and equate coefficients: \(\quad\mathbf{i}\colon\ -4 = -2 + 5v_1 \qquad v_1 = -\tfrac{2}{5}\) |
| \(\mathbf{j}\colon\ 17 = 5 + 5v_2 \qquad v_2 = \tfrac{12}{5}\) |
| Velocity of \(B\) after collision \(= -\tfrac{2}{5}\mathbf{i} + \tfrac{12}{5}\mathbf{j}\). |