A2 October 2021 Q3
3. [In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane.]
A smooth uniform sphere \(P\) has mass 0.3 kg. Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.5 kg.
The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \((u\mathbf{i} + 2\mathbf{j})\ \text{m s}^{-1}\), where \(u\) is a positive constant, and the velocity of \(Q\) is \((-4\mathbf{i} + 3\mathbf{j})\ \text{m s}^{-1}\)
At the instant when the spheres collide, the line joining their centres is parallel to \(\mathbf{i}\).
The coefficient of restitution between \(P\) and \(Q\) is \(\dfrac{3}{5}\)
As a result of the collision, the direction of motion of \(P\) is deflected through an angle of \(90^\circ\) and the direction of motion of \(Q\) is deflected through an angle of \(\alpha^\circ\)
| Scheme | Marks | AO |
|---|---|---|
 | ||
| For \(P\) after: Component in \(\mathbf{j}\) direction \(= 2\) | B1 | 3.4 |
| Deflected through \(90^\circ\) so velocity after \(= \left(-\dfrac{4}{u}\mathbf{i} + 2\mathbf{j}\right)\ (\text{m s}^{-1})\) | B1 | 3.4 |
| CLM parallel to line of centres: | M1 | 3.1a |
| \(0.3\left(u + \dfrac{4}{u}\right) = 0.5(4 + w)\) | A1ft | 1.1b |
| Impact law parallel to line of centres: | M1 | 3.1a |
| \(w + \dfrac{4}{u} = \dfrac{3}{5}(u + 4)\) | A1ft | 1.1b |
| \(\left\{\begin{aligned} 3u + \dfrac{12}{u} &= 20 + 5w \\ \dfrac{20}{u} + 5w &= 3u + 12 \end{aligned}\right.\) | M1 | 1.1b |
| \(\left(\Rightarrow \dfrac{32}{u} = 32,\right)\ \ u = 1\) | A1 | 2.2a |
| (8) |
Notes
B1: Correct only Check the diagram
B1: Correct only. Seen or implied.
M1: Correct use of CLM. Need all terms. Condone sign errors
A1ft: Follow their components of velocity of \(P\), with or without a value for the \(\mathbf{i}\) component.
M1: Correct use of the impact law. Condone sign errors
A1ft: Follow their components of velocity of \(P\), with or without a value for the \(\mathbf{i}\) component.
M1: Solve their correctly formed simultaneous equations to obtain value of \(u\).
A1: Correct only
| Scheme | Marks | AO |
|---|---|---|
| For \(Q\) after: \(w = -1\) | B1 | 1.1b |
| \(\mathbf{v} = -\mathbf{i} + 3\mathbf{j}\) | B1ft | 1.1b |
| Find relevant angle between directions | M1 | 3.1a |
| \(\alpha^\circ = \tan^{-1}3 - \tan^{-1}\dfrac{3}{4}\) or \(\alpha^\circ = \cos^{-1}\left(\dfrac{4 + 9}{5 \times \sqrt{10}}\right)\) | A1ft | 1.1b |
| \(\alpha = 34.7\ \ (35)\) | A1 | 1.1b |
| (5) |
Notes
B1: Correct only
B1ft: Follow their \(w\)
M1: Correct method to find a relevant angle between the directions
A1ft: Correct unsimplified expression. Follow their \(\mathbf{v}\)
A1: 35 or better (34.695…) 0.61 radians
| Scheme | Marks | AO |
|---|---|---|
| The line of centres is parallel to the surface the spheres are moving on, so the impulse acts parallel to the surface. | B1 | 3.5a |
| (1) | ||
| (14 marks) |
Notes
B1: Or equivalent that explains that the line of centres is parallel to the surface