M4 June 2017 Q2
2.
Two smooth uniform spheres \(A\) and \(B\) have masses \(3m\) kg and \(m\) kg respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \((5\mathbf{i} - 2\mathbf{j})\) m s\(^{-1}\) and sphere \(B\) has velocity \((3\mathbf{i} + 4\mathbf{j})\) 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 two spheres is \(e\).
The kinetic energy of sphere \(B\) immediately after the collision is 85% of its kinetic energy immediately before the collision.
Find
| Scheme | Marks |
|---|---|
| For \(A\), component perpendicular to loc = 5 | B1 |
| For \(B\), component perpendicular to loc = 3 | B1 |
| \(\dfrac{1}{2}m\times 25\times\dfrac{85}{100} = \dfrac{1}{2}m\left(3^2 + v^2\right)\) | M1 |
| \(\dfrac{85}{4} = 9 + v^2,\quad v^2 = \dfrac{49}{4}\) | A1 |
| \(-6m + 4m = 3mw - mv\) \((= 3mw - 3.5m)\) | M1 |
| A1ft | |
| \(w = 0.5\) | |
| Select correct root and state velocities: | DM1 |
| \(\mathbf{v}_B = (3\mathbf{i} - 3.5\mathbf{j})\) (m s\(^{-1}\)) | A1 |
| \(\mathbf{v}_A = (5\mathbf{i} + 0.5\mathbf{j})\) (m s\(^{-1}\)) | A1 |
| (9) |
Notes
M1 Equation for kinetic energy of \(B\). For their "3"
M1 CLM parallel to loc. No missing/additional terms. Condone sign error(s)
A1ft Correct unsimplified equation for CLM (with their values if substituted)
A1 One correct
A1 Both correct
(Corrected from the printed mark scheme: the kinetic energy equation is printed with \(3^3\) for \(3^2\).)
| Scheme | Marks |
|---|---|
| \(v + w = e(2 + 4)\) | M1 |
| \(0.5 + 3.5 = 6e\) | A1ft |
| \(e = \dfrac{2}{3}\) | A1 |
| (3) | |
| (12 marks) |
Notes
M1 Impact law parallel to loc. Used the right way round. Condone sign error(s)
A1ft Correct unsimplified or with their values