M4 June 2007 Q5
5. A smooth uniform sphere \(A\) has mass \(2m\) kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(m\) kg. The spheres are moving on a smooth horizontal plane when they collide. At the instant of collision the line joining the centres of the spheres is parallel to \(\mathbf{j}\). Immediately after the collision, the velocity of \(A\) is \((3\mathbf{i} - \mathbf{j})\) m s\(^{-1}\) and the velocity of \(B\) is \((2\mathbf{i} + \mathbf{j})\) m s\(^{-1}\). The coefficient of restitution between the spheres is \(\tfrac{1}{2}\).
| Scheme | Marks |
|---|---|
| CLM: \(\quad 2v_2 - v_1 = 1 - 2 = -1\) | M1A1 |
| NIL: \(\quad 1 + 1 = \dfrac{1}{2}(v_1 + v_2)\) | M1A1 |
| \(\therefore v_2 = 1,\ v_1 = 3\qquad\) Dependent on both M’s above | DM1 |
| Horizontal components unchanged (i.e. 2 & 3)\(\qquad\) Independent of all other marks | A1 |
| \(\mathbf{v}_A = 3\mathbf{i} + \mathbf{j};\ \mathbf{v}_B = 2\mathbf{i} - 3\mathbf{j}\) | A1 |
| (7) |
Notes
M1 Conservation of momentum along the line of centres. Condone sign errors
A1 equation correct
M1 Impact law along the line of centres. \(e\) must be used correctly, but condone sign errors.
A1 equation correct. The signs need to be consistent between the two equations
M1 Solve the simultaneous equations for their v1 and v2.
A1 \(\mathbf{i}\) components correct – independent mark
A1 \(\mathbf{v}_A\) & \(\mathbf{v}_B\) correct
Special case
Special case: candidates who act as if the line of centres is in the direction of \(\mathbf{i}\):
CLM u+2v = 8
NIL v-u = 2
u=4/3, v=10/3
4/3i + j ; 10/3i – j
Impulse 2m-4/3m = 2/3m
\(\dfrac{10 + 1}{\sqrt{10}\sqrt{\dfrac{109}{9}}} = \cos\theta\qquad \theta = 1.70^0\)
Work is equivalent, so treat as a MR:
M1A0M1A0M1A1A1 M1A1 M1A1M1A1
| Scheme | Marks |
|---|---|
| For B: I = m(1-(-3)) = 4m | M1A1 |
| (Or For A: -I = 2m(-1 – 1) \(\ \therefore\) I = 4m) | |
| (2) |
Notes
M1 Impulse = change in momentum for one sphere. Condone order of subtraction.
A1 Magnitude correct.
| Scheme | Marks |
|---|---|
| \(\begin{pmatrix}3\\1\end{pmatrix}.\begin{pmatrix}3\\-1\end{pmatrix} = \sqrt{3^2 + 1^2}.\sqrt{3^2 + (-1)^2}\cos\theta\) | M1A1 |
| \(\Rightarrow 8 = 10\cos\theta\) | M1 |
| \(\theta = 37^o\) | A1 |
| (4) | |
| (13 marks) |
Notes
Alternative
| M1 | |
| where \(\tan\theta = \dfrac{1}{3}\) | A1 |
| required angle is \(2\theta\) | M1A1 |
M1 Any complete method to find the trig ratio of a relevant angle.
A1 \(\cos\theta = \dfrac{4}{5},\ \tan\dfrac{\theta}{2} = \dfrac{1}{3},\ \ldots\)
Or M1 find angle of approach to the line of centres and angle after collision.
A1 values correct. (both 71.56 …..)
M1 solve for \(\theta\)
A1 \(37^0\) (Q specifies nearest degree)