| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2022 |
| Session | October |
| 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 M2 collision problem with multiple stages (particle collision, wall rebound, second collision). Part (a) uses routine momentum conservation and restitution formula with algebraic manipulation shown. Part (b) requires tracking velocities through the wall bounce and using kinematics to find meeting time. While multi-step, all techniques are standard M2 fare with no novel insight required, making it slightly easier than average. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Conservation of momentum | M1 | Need all terms. Dimensionally correct. Condone sign errors. Condone \(m\) missing throughout or \(g\) present throughout |
| \(3mu + 2mu = mv_A + 2mv_B \quad (5u = v_A + 2v_B)\) | A1 | Correct unsimplified equation. Allow with \(v_A\) negative |
| Use of NEL | M1 | Used the right way round. Condone sign errors |
| \(v_B - v_A = e(3u - u) \quad (2ue = v_B - v_A)\) | A1 | Correct unsimplified equation. Allow with \(v_A\) negative. Signs consistent between the two equations |
| Solve for \(v_A\) or \(v_B\) | DM1 | Dependent on two previous M marks |
| \(v_B = \frac{5+2e}{3}u\) * | A1* | Obtain given answer from correct working |
| \(v_A = \frac{5-4e}{3}u\) | A1 | Or equivalent. \(v_A\) must be positive |
| Total | 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Time for \(B\) to reach the wall \(t_B = \frac{d}{2u}\) | B1 | Seen or implied. Allow \(\frac{d \times 3}{(5+2e)u}\) |
| Speed of \(B\) after impact with wall \(= \frac{2}{3}u\) | B1 | Seen or implied. Allow \(\frac{1}{3}\left(\frac{5+2e}{3}\right)\) |
| Distance travelled by \(A\) before \(B\) hits the wall \(= u \times \frac{d}{2u}\left(= \frac{d}{2}\right)\) | M1 | Substitute \(e = \frac{1}{2}\) and use their \(v_A\) and their \(t_b\) to find distance |
| Time to close the gap | M1 | Correct formula with their relevant speeds |
| \(\frac{d}{2} = u \times t + \frac{2u}{3} \times t \left(= \frac{5ut}{3}\right) \quad \left(t = \frac{3d}{10u}\right)\) | A1 | Correct unsimplified equation |
| Total time \(= \frac{d}{2u} + \frac{3d}{10u} = \frac{8d}{10u}\left(= \frac{4d}{5u}\right)\) | A1 | ISW. Any equivalent form |
| Alt 7(b): In time \(T\), \(A\) travels \(x\) metres \(x = uT\); \(B\) travels \(d\) metres in \(t\) sec, \(d = 2ut\) (First B1); \(B\) travels \(d-x\) metres in \(t'\) sec, \(d-x = 2ut'/3\) (Second B1 and first M1); \(t + t' = T\); \((d + 3d - 3uT)/2u = T\) (Second M1 and first A1); \(T = 4d/5u\) (Second A1) | Equivalent statements throughout | |
| Total | 6 |
# Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Conservation of momentum | M1 | Need all terms. Dimensionally correct. Condone sign errors. Condone $m$ missing throughout or $g$ present throughout |
| $3mu + 2mu = mv_A + 2mv_B \quad (5u = v_A + 2v_B)$ | A1 | Correct unsimplified equation. Allow with $v_A$ negative |
| Use of NEL | M1 | Used the right way round. Condone sign errors |
| $v_B - v_A = e(3u - u) \quad (2ue = v_B - v_A)$ | A1 | Correct unsimplified equation. Allow with $v_A$ negative. Signs consistent between the two equations |
| Solve for $v_A$ or $v_B$ | DM1 | Dependent on two previous M marks |
| $v_B = \frac{5+2e}{3}u$ * | A1* | Obtain **given answer** from correct working |
| $v_A = \frac{5-4e}{3}u$ | A1 | Or equivalent. $v_A$ must be positive |
| **Total** | **7** | |
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# Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Time for $B$ to reach the wall $t_B = \frac{d}{2u}$ | B1 | Seen or implied. Allow $\frac{d \times 3}{(5+2e)u}$ |
| Speed of $B$ after impact with wall $= \frac{2}{3}u$ | B1 | Seen or implied. Allow $\frac{1}{3}\left(\frac{5+2e}{3}\right)$ |
| Distance travelled by $A$ before $B$ hits the wall $= u \times \frac{d}{2u}\left(= \frac{d}{2}\right)$ | M1 | Substitute $e = \frac{1}{2}$ and use their $v_A$ and their $t_b$ to find distance |
| Time to close the gap | M1 | Correct formula with their relevant speeds |
| $\frac{d}{2} = u \times t + \frac{2u}{3} \times t \left(= \frac{5ut}{3}\right) \quad \left(t = \frac{3d}{10u}\right)$ | A1 | Correct unsimplified equation |
| Total time $= \frac{d}{2u} + \frac{3d}{10u} = \frac{8d}{10u}\left(= \frac{4d}{5u}\right)$ | A1 | ISW. Any equivalent form |
| **Alt 7(b):** In time $T$, $A$ travels $x$ metres $x = uT$; $B$ travels $d$ metres in $t$ sec, $d = 2ut$ (First B1); $B$ travels $d-x$ metres in $t'$ sec, $d-x = 2ut'/3$ (Second B1 and first M1); $t + t' = T$; $(d + 3d - 3uT)/2u = T$ (Second M1 and first A1); $T = 4d/5u$ (Second A1) | | Equivalent statements throughout |
| **Total** | **6** | |
---7. Particle $A$ has mass $m$ and particle $B$ has mass $2 m$.
The particles are moving in the same direction along the same straight line on a smooth horizontal surface.\\
Particle $A$ collides directly with particle $B$.\\
Immediately before the collision, the speed of $A$ is $3 u$ and the speed of $B$ is $u$.\\
The coefficient of restitution between $A$ and $B$ is $e$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that the speed of $B$ immediately after the collision is $\frac { 5 + 2 e } { 3 } u$
\item Find the speed of $A$ immediately after the collision.
After the collision, $B$ hits a smooth fixed vertical wall that is perpendicular to the direction of motion of $B$.\\
The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 3 }$\\
Particle $B$ rebounds and there is a second collision between $A$ and $B$.\\
The first collision between $A$ and $B$ occurs at a distance $d$ from the wall.\\
The time between the two collisions is $T$.\\
Given that $e = \frac { 1 } { 2 }$
\end{enumerate}\item find $T$ in terms of $d$ and $u$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2022 Q7 [13]}}