| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2004 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Two-sphere oblique collision |
| Difficulty | Challenging +1.2 This is a multi-step M4 collision problem requiring work-energy theorem to find post-collision speeds, then conservation of momentum in 2D, and finally coefficient of restitution. While it involves several techniques and careful bookkeeping across multiple parts, the individual steps are standard M4 procedures without requiring novel insight or particularly complex mathematical manipulation. |
| Spec | 3.02d Constant acceleration: SUVAT formulae6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03d Conservation in 2D: vector momentum6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| LM \(600u = 800x\) | M1, A1 | |
| NEL \(x = eu\) | M1, A1 | |
| \(e = 0.75\) | A1 | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Van N2L \(-500 = 800a\) | M1 | |
| \(0^2 = x^2 - 2 \times 0.625 \times 45\), \(x^2 = 56.25\) (\(x = 7.5\)) | M1, A1 | |
| Car N2L \(-300 = 600a\) | M1 | |
| \(0^2 = v^2 - 2 \times 0.5 \times 21\), \(v^2 = 21\) | M1, A1 | |
| From (a) NEL \(u = \frac{4}{3} \times 7.5 = 10\) | M1 | |
| \(V^2 = 10^2 + 21 \Rightarrow V = 11\) (ms\(^{-1}\)) | M1, A1 (cao) | (9 marks) |
## Part (a)
**LM** $600u = 800x$ | M1, A1 |
**NEL** $x = eu$ | M1, A1 |
$e = 0.75$ | A1 | (5 marks)
## Part (b)
**Van N2L** $-500 = 800a$ | M1 |
$0^2 = x^2 - 2 \times 0.625 \times 45$, $x^2 = 56.25$ ($x = 7.5$) | M1, A1 |
**Car N2L** $-300 = 600a$ | M1 |
$0^2 = v^2 - 2 \times 0.5 \times 21$, $v^2 = 21$ | M1, A1 |
From (a) **NEL** $u = \frac{4}{3} \times 7.5 = 10$ | M1 |
$V^2 = 10^2 + 21 \Rightarrow V = 11$ (ms$^{-1}$) | M1, A1 (cao) | (9 marks)
**Total: 14 marks**
---\includegraphics{figure_3}
Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point $X$ with a stationary van of mass 800 kg. After the collision the van came to rest at the point $A$ having travelled a horizontal distance of 45 m, and the car came to rest at the point $B$ having travelled a horizontal distance of 21 m. The angle $AXB$ is 90°.
The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.
\begin{enumerate}[label=(\alph*)]
\item Find the coefficient of restitution between the car and the van.
[5]
\end{enumerate}
The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along $AX$ and the car to a constant horizontal force of 300 N along $BX$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the speed of the car immediately before the collision.
[9]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2004 Q5 [14]}}