| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2012 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, direction deflected given angle |
| Difficulty | Challenging +1.2 This is a standard M4 oblique collision problem requiring resolution of velocities along/perpendicular to the line of centres, application of conservation of momentum and Newton's restitution law. While it involves multiple steps and careful bookkeeping of components, the techniques are routine for this module with no novel insight required. The given tan α = 3/4 simplifies trigonometry. Slightly above average difficulty due to the algebraic manipulation and multi-part nature, but well within standard M4 expectations. |
| Spec | 6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| \(mu\cos\alpha = mw + 2mV\) | M1 A1 | CLM parallel to the line of centres. \(\frac{4}{5}u = w + 2V\). Need all terms but condone sign errors. |
| \(eu\cos\alpha = -w + V\) | M1 A1 | Impact law. Must be the right way round. \(\frac{3}{4} \times \frac{4}{5}u = V - w\) |
| \(u\cos\alpha(e+1) = 3V \Rightarrow (i) u = \frac{15V}{7}\) | M1 A1 | Eliminate \(w\) and solve for \(u\) in terms of \(V\) or v.v. 2.14V or better |
| \(\Rightarrow w = -\frac{2V}{7}\) | A1 | Solve for \(w\) in terms of \(V\). -0.286V or better |
| \((ii)\) speed of \(S = \sqrt{\left(\frac{-2V}{7}\right)^2 + (u\sin\alpha)^2} = \frac{V\sqrt{85}}{7}\) | M1 A1 | Use of Pythagoras with their \(u\sin\alpha\) and \(w\). \(\sqrt{\left(\frac{-2V}{7}\right)^2 + \left(\frac{15V}{7} \times \frac{3}{5}\right)^2}\) \(\sqrt{\frac{85}{49}}V\), accept 1.32V or better |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan\theta = \frac{9V}{7} \div \frac{2V}{7} = \frac{9}{2}\) | M1 A1 | Direction of \(S\) after the collision. Condone \(\frac{2}{9}\) 77.5° or 12.5° seen or implied. Combine their \(\theta\) and \(\alpha\) to find the required angle. e.g. \(12.5° + \tan^{-1}\left(\frac{4}{3}\right)\) |
| defln angle \(= 180° - (\theta + \alpha) = 65.7°\) (3 sf) | DM1 A1 | Set their derivative = 0. First answer. Accept 66° |
**Part (a)**
| $mu\cos\alpha = mw + 2mV$ | M1 A1 | CLM parallel to the line of centres. $\frac{4}{5}u = w + 2V$. Need all terms but condone sign errors. |
| $eu\cos\alpha = -w + V$ | M1 A1 | Impact law. Must be the right way round. $\frac{3}{4} \times \frac{4}{5}u = V - w$ |
| $u\cos\alpha(e+1) = 3V \Rightarrow (i) u = \frac{15V}{7}$ | M1 A1 | Eliminate $w$ and solve for $u$ in terms of $V$ or v.v. 2.14V or better |
| $\Rightarrow w = -\frac{2V}{7}$ | A1 | Solve for $w$ in terms of $V$. -0.286V or better |
| $(ii)$ speed of $S = \sqrt{\left(\frac{-2V}{7}\right)^2 + (u\sin\alpha)^2} = \frac{V\sqrt{85}}{7}$ | M1 A1 | Use of Pythagoras with their $u\sin\alpha$ and $w$. $\sqrt{\left(\frac{-2V}{7}\right)^2 + \left(\frac{15V}{7} \times \frac{3}{5}\right)^2}$ $\sqrt{\frac{85}{49}}V$, accept 1.32V or better |
**Part (b)**
| $\tan\theta = \frac{9V}{7} \div \frac{2V}{7} = \frac{9}{2}$ | M1 A1 | Direction of $S$ after the collision. Condone $\frac{2}{9}$ 77.5° or 12.5° seen or implied. Combine their $\theta$ and $\alpha$ to find the required angle. e.g. $12.5° + \tan^{-1}\left(\frac{4}{3}\right)$ |
| defln angle $= 180° - (\theta + \alpha) = 65.7°$ (3 sf) | DM1 A1 | Set their derivative = 0. First answer. Accept 66° |
**(9) Total for Question 1: (4) 13 marks**
---\begin{enumerate}
\item A smooth uniform sphere $S$, of mass $m$, is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere $T$, of the same radius as $S$ but of mass $2 m$, which is at rest on the plane. Immediately before the collision the velocity of $S$ makes an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, with the line joining the centres of the spheres. Immediately after the collision the speed of $T$ is $V$. The coefficient of restitution between the spheres is $\frac { 3 } { 4 }$.\\
(a) Find, in terms of $V$, the speed of $S$\\
(i) immediately before the collision,\\
(ii) immediately after the collision.\\
(b) Find the angle through which the direction of motion of $S$ is deflected as a result of the collision.\\
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2012 Q1 [13]}}