| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2009 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.3 This is a standard M3 collision problem requiring systematic application of conservation of momentum and Newton's restitution law across multiple collision events. While it has three parts and requires careful algebraic manipulation, the techniques are routine for this module—no novel insight is needed, just methodical application of standard formulas. The 'show that' format guides students to the answer, reducing problem-solving demand. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(5mu + 7mu = mv_A + 7mv_B\) | M1A1 | Allow consistent use of positive or negative sign for \(v_A\) |
| \(12u = v_A + 7v_B\) | ||
| \(e = \dfrac{-v_A + v_B}{4u}\) | M1 | |
| \(-v_A + v_B = 4eu\) | ||
| \(8v_B = 12u + 4eu\) | m1 | |
| \(v_B = \dfrac{u}{2}(e+3)\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(v_A = \dfrac{u}{2}(e+3) - 4eu\) | M1 | |
| \(v_A = \dfrac{u}{2}(3-7e)\) | A1F | |
| \(\dfrac{u}{2}(3-7e) < 0\) | M1 | |
| \(3 - 7e < 0\) | ||
| \(e > \dfrac{3}{7}\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(w_B = \dfrac{u}{4}(e+3)\) | M1 | |
| \(\dfrac{u}{2}(7e-3) < \dfrac{u}{4}(e+3)\) | M1 | |
| \(2(7e-3) < e+3\) | ||
| \(13e < 9\) | m1 | |
| \(e < \dfrac{9}{13}\) | A1 | AG |
## Question 6:
### Part (a):
| Working | Marks | Guidance |
|---------|-------|----------|
| $5mu + 7mu = mv_A + 7mv_B$ | M1A1 | Allow consistent use of positive or negative sign for $v_A$ |
| $12u = v_A + 7v_B$ | | |
| $e = \dfrac{-v_A + v_B}{4u}$ | M1 | |
| $-v_A + v_B = 4eu$ | | |
| $8v_B = 12u + 4eu$ | m1 | |
| $v_B = \dfrac{u}{2}(e+3)$ | A1 | AG |
### Part (b):
| Working | Marks | Guidance |
|---------|-------|----------|
| $v_A = \dfrac{u}{2}(e+3) - 4eu$ | M1 | |
| $v_A = \dfrac{u}{2}(3-7e)$ | A1F | |
| $\dfrac{u}{2}(3-7e) < 0$ | M1 | |
| $3 - 7e < 0$ | | |
| $e > \dfrac{3}{7}$ | A1 | AG |
### Part (c):
| Working | Marks | Guidance |
|---------|-------|----------|
| $w_B = \dfrac{u}{4}(e+3)$ | M1 | |
| $\dfrac{u}{2}(7e-3) < \dfrac{u}{4}(e+3)$ | M1 | |
| $2(7e-3) < e+3$ | | |
| $13e < 9$ | m1 | |
| $e < \dfrac{9}{13}$ | A1 | AG |
---6 A smooth sphere $A$ of mass $m$ is moving with speed $5 u$ in a straight line on a smooth horizontal table. The sphere $A$ collides directly with a smooth sphere $B$ of mass $7 m$, having the same radius as $A$ and moving with speed $u$ in the same direction as $A$. The coefficient of restitution between $A$ and $B$ is $e$.\\
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-5_287_880_529_571}
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $B$ after the collision is $\frac { u } { 2 } ( e + 3 )$.
\item Given that the direction of motion of $A$ is reversed by the collision, show that $e > \frac { 3 } { 7 }$.
\item Subsequently, $B$ hits a wall fixed at right angles to the direction of motion of $A$ and $B$. The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 2 }$. Given that after $B$ rebounds from the wall both spheres move in the same direction and collide again, show also that $e < \frac { 9 } { 13 }$.\\
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2009 Q6 [13]}}