| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, find velocities/angles |
| Difficulty | Challenging +1.2 This is a standard oblique collision problem requiring systematic application of conservation of momentum (parallel and perpendicular to line of centres) and Newton's restitution law. While it involves multiple unknowns and algebraic manipulation across several equations, the approach is methodical and follows textbook procedures for M3. The multi-part structure and need to handle components in two directions elevates it slightly above average difficulty, but it requires no novel insight beyond standard collision mechanics. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03l Newton's law: oblique impacts |
| Answer | Marks |
|---|---|
| M1* | must be 3 non-zero terms |
| A1 | allow sign errors, \(m\)/2\(m\) errors, \(\sin/\cos\) |
| M1* | must be 3 non-zero terms, and '\(e\)' in correct position |
| A1 | allow sign errors, \(\sin/\cos\) |
| *M1 | dep both previous M1 marks |
| A1 | (AG) |
| A1 [7] | dep final M1 and www |
| Answer | Marks |
|---|---|
| B1 | |
| M1 | need to eliminate \(a\) or \(\alpha\) |
| A1 | |
| A1 [4] | accept 1.03 radians |
### Part (i)
**Answer:** use of conservation of momentum
$2m\cos\alpha - mb\cos\beta = mx2x\cos45°$
use of NEL
2$\cos 45° - 0 = -2/3 (-b\cos\beta - a\cos\alpha)$
attempt to eliminate $a\cos\alpha$ or $b\cos\beta$
$a\cos\alpha = 5\sqrt{2}/6$
$b\cos\beta = 2\sqrt{2}/3$ oe
| M1* | must be 3 non-zero terms |
| A1 | allow sign errors, $m$/2$m$ errors, $\sin/\cos$ |
| M1* | must be 3 non-zero terms, and '$e$' in correct position |
| A1 | allow sign errors, $\sin/\cos$ |
| *M1 | dep both previous M1 marks |
| A1 | (AG) |
| A1 [7] | dep final M1 and www |
### Part (ii)
**Answer:** $a\sin\alpha = 2$
attempt to solve $a\sin\alpha = 2$ and $a\cos\alpha = 5\sqrt{2}/6$
$a = 2.32$
$\alpha = 59.5°$
| B1 | |
| M1 | need to eliminate $a$ or $\alpha$ |
| A1 | |
| A1 [4] | accept 1.03 radians |
**Guidance:** 2.321398..., 59.49104...°, 1.0383...rad\includegraphics{figure_5}
Two uniform smooth spheres $A$ and $B$, of equal radius, have masses $2m$ kg and $m$ kg respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, $A$ is moving with speed $a\text{ ms}^{-1}$ in a direction making an angle $\alpha$ with the line of centres and $B$ is moving towards $A$ with speed $b\text{ ms}^{-1}$ in a direction making an angle $\beta$ with the line of centres (see diagram). After the collision, $A$ moves with velocity $2\text{ ms}^{-1}$ in a direction perpendicular to the line of centres and $B$ moves with velocity $2\text{ ms}^{-1}$ in a direction making an angle of $45°$ with the line of centres. The coefficient of restitution between $A$ and $B$ is $\frac{2}{3}$.
\begin{enumerate}[label=(\roman*)]
\item Show that $a\cos \alpha = \frac{5}{3}\sqrt{2}$ and find $b\cos \beta$. [7]
\item Find the values of $a$ and $\alpha$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR M3 2015 Q5 [11]}}