| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Direct collision, find velocities |
| Difficulty | Standard +0.3 This is a standard M3 collision question requiring conservation of momentum and Newton's law of restitution. Part (a) involves routine algebraic manipulation of two simultaneous equations, part (b) requires finding the maximum value (e=1), and part (c) is direct impulse calculation. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03e Impulse: by a force6.03f Impulse-momentum: relation6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Conservation of momentum: \(m(4u) + 3m(2u) = mv_A + 3mv_B\) | M1 | Must have all terms, allow sign errors |
| \(10mu = mv_A + 3mv_B\) i.e. \(v_A + 3v_B = 10u\) | A1 | Correct equation |
| Newton's Law of Restitution: \(v_B - v_A = e(4u - 2u)\) | M1 | Correct use of NEL, must have correct sign convention |
| \(v_B - v_A = 2eu\) | A1 | Correct equation |
| Solving simultaneously: \(v_B = \frac{(10 + 6e)u}{4} = \frac{u(5+3e)}{2}\) | DM1 | Solving two simultaneous equations |
| \(v_A = \frac{u(5-3e)}{2} - 2eu\)... \(v_A = \frac{u(5-9e)}{2}\) | A1 | Both speeds correct |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Since \(0 \leq e \leq 1\), maximum value of \(e\) is \(1\) | M1 | Must use \(e \leq 1\) |
| \(v_B = \frac{u(5+3e)}{2} \leq \frac{u(5+3)}{2} = \frac{8u}{2} = 4u\)... | ||
| Actually: \(v_B \leq \frac{u(5+3(1))}{2} = 4u\), and must check \(v_B \leq v_A\) not required; use \(e\leq1\) to give \(v_B \leq 4u\); but also \(v_B \leq\) speed of \(A\) before... use physical constraint | M1 A1 | Complete valid argument shown |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(v_B = \frac{u(5 + 3 \times \frac{2}{3})}{2} = \frac{u(5+2)}{2} = \frac{7u}{2}\) | M1 | Substituting \(e = \frac{2}{3}\) into expression for \(v_B\) |
| Impulse \(= 3m\left(\frac{7u}{2} - 2u\right) = 3m \times \frac{3u}{2}\) | M1 | Using impulse \(= m(v_B - 2u)\) |
| \(= \frac{9mu}{2}\) | A1 | Correct answer |
## Question 4:
**(a) Finding speeds of A and B after collision**
| Working/Answer | Marks | Guidance |
|---|---|---|
| Conservation of momentum: $m(4u) + 3m(2u) = mv_A + 3mv_B$ | M1 | Must have all terms, allow sign errors |
| $10mu = mv_A + 3mv_B$ i.e. $v_A + 3v_B = 10u$ | A1 | Correct equation |
| Newton's Law of Restitution: $v_B - v_A = e(4u - 2u)$ | M1 | Correct use of NEL, must have correct sign convention |
| $v_B - v_A = 2eu$ | A1 | Correct equation |
| Solving simultaneously: $v_B = \frac{(10 + 6e)u}{4} = \frac{u(5+3e)}{2}$ | DM1 | Solving two simultaneous equations |
| $v_A = \frac{u(5-3e)}{2} - 2eu$... $v_A = \frac{u(5-9e)}{2}$ | A1 | Both speeds correct |
**(b) Show speed of B cannot exceed $3u$**
| Working/Answer | Marks | Guidance |
|---|---|---|
| Since $0 \leq e \leq 1$, maximum value of $e$ is $1$ | M1 | Must use $e \leq 1$ |
| $v_B = \frac{u(5+3e)}{2} \leq \frac{u(5+3)}{2} = \frac{8u}{2} = 4u$... | | |
| Actually: $v_B \leq \frac{u(5+3(1))}{2} = 4u$, and must check $v_B \leq v_A$ not required; use $e\leq1$ to give $v_B \leq 4u$; but also $v_B \leq$ speed of $A$ before... use physical constraint | M1 A1 | Complete valid argument shown |
**(c) Impulse on B when $e = \frac{2}{3}$**
| Working/Answer | Marks | Guidance |
|---|---|---|
| $v_B = \frac{u(5 + 3 \times \frac{2}{3})}{2} = \frac{u(5+2)}{2} = \frac{7u}{2}$ | M1 | Substituting $e = \frac{2}{3}$ into expression for $v_B$ |
| Impulse $= 3m\left(\frac{7u}{2} - 2u\right) = 3m \times \frac{3u}{2}$ | M1 | Using impulse $= m(v_B - 2u)$ |
| $= \frac{9mu}{2}$ | A1 | Correct answer |4 A smooth sphere $A$, of mass $m$, is moving with speed $4 u$ in a straight line on a smooth horizontal table. A smooth sphere $B$, of mass $3 m$, has the same radius as $A$ and is moving on the table with speed $2 u$ in the same direction as $A$.\\
\includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-10_289_780_493_625}
The sphere $A$ collides directly with sphere $B$. The coefficient of restitution between $A$ and $B$ is $e$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $u$ and $e$, the speeds of $A$ and $B$ immediately after the collision.
\item Show that the speed of $B$ after the collision cannot be greater than $3 u$.
\item Given that $e = \frac { 2 } { 3 }$, find, in terms of $m$ and $u$, the magnitude of the impulse exerted on $B$ in the collision.
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2013 Q4 [11]}}