M4 June 2011 Q1
1.
Two smooth uniform spheres \(A\) and \(B\) have masses \(2m\) kg and \(3m\) kg respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \((3\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\) and sphere \(B\) has velocity \((2\mathbf{i} - 3\mathbf{j})\) m s\(^{-1}\). When the spheres collide, the line joining their centres is parallel to \(\mathbf{j}\), as shown in Figure 1. The coefficient of restitution between the spheres is \(\dfrac{3}{7}\). Find, in terms of \(m\), the total kinetic energy lost in the collision. (10)
| Scheme | Marks |
|---|---|
 | |
| \(\leftrightarrow a = 3\ \&\ b = 2\) | B1 |
| b Conservation of linear momentum: \(\ -4 \times 2 + 3 \times 3 = 2v - 3w\ (= 1)\) | M1 A1 |
| Restitution: \(\ v + w = e \times 7\ \ (= 3)\) | M1 A1 |
| Solve the simultaneous equations | DM1 |
| giving \(v = 2\) and \(w = 1\) | A1 |
| KE lost \(= \dfrac{1}{2} \times 2m \times ((16 + 9) - (4 + 9)) + \dfrac{1}{2} \times 3m \times ((9 + 4) - (1 + 4))\) | M1 A1 |
| \(= 24m\) (J) | A1 |
| (10 marks) |
Notes
(Corrected from the printed mark scheme: the KE lost line is printed as \(\frac{1}{2} \times 2m \times ((16 + 9) - (4 - 9)) + \frac{1}{2} \times 3m \times ((9 + 4) - (1 - 4))\); the speeds squared after the collision are \(4 + 9\) and \(1 + 4\), which give the printed answer \(24m\).)
(The question text gives the velocity of \(B\) as \((2\mathbf{i} - 3\mathbf{j})\) m s\(^{-1}\); Figure 1 and this mark scheme use \((2\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\).)