| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2018 |
| Session | January |
| 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 two stages: particle-particle collision using conservation of momentum and Newton's restitution law, followed by a wall collision. Part (a) requires routine application of collision formulas, part (b) is a straightforward inequality proof, and part (c) involves finding conditions for a second collision. While multi-step, all techniques are standard M2 material with no novel insight required, making it slightly easier than average. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Conservation of momentum | M1 | Dimensionally correct. All terms required. Condone sign errors |
| \(2mu - 3mu = -2mv_P + mv_Q\ (-u = -2v_P + v_Q)\) | A1 | Correct unsimplified equation |
| Impact law | M1 | Must be used the right way round. Condone sign errors. |
| \(4eu = v_P + v_Q\) | A1 | Correct unsimplified equation. Signs consistent with CLM equation |
| Solve simultaneous equations for \(v_P\) or \(v_Q\) | DM1 | Dependent on the 2 preceding M marks |
| \(v_P = \frac{u(1+4e)}{3}\) | A1 | One correct (must be positive) or equivalent |
| \(v_Q = \frac{u(8e-1)}{3}\) | A1 | Both correct (must be positive) or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{u(1+4e)}{3} > 0\) | M1 | Working from a correct inequality for their \(v_P\) |
| Always true because \(e \geq 0\) \(\left(\text{or } e > \frac{1}{8}\right)\) | A1 | Correct justification from correct work |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(e = \frac{3}{4} \Rightarrow v_Q = \frac{5u}{3}\) | B1 | Seen or implied |
| Speed of \(Q\) after collision \(= fv_Q\) | M1 | Impact law for collision with the wall |
| To collide with \(P\): \(fv_2 > v_1 = \frac{4u}{3}\) | M1 | Correct inequality for second collision |
| \(1 \geq f > \frac{4}{5}\) | A1 | Both limits required |
## Question 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Conservation of momentum | M1 | Dimensionally correct. All terms required. Condone sign errors |
| $2mu - 3mu = -2mv_P + mv_Q\ (-u = -2v_P + v_Q)$ | A1 | Correct unsimplified equation |
| Impact law | M1 | Must be used the right way round. Condone sign errors. |
| $4eu = v_P + v_Q$ | A1 | Correct unsimplified equation. Signs consistent with CLM equation |
| Solve simultaneous equations for $v_P$ or $v_Q$ | DM1 | Dependent on the 2 preceding M marks |
| $v_P = \frac{u(1+4e)}{3}$ | A1 | One correct (must be positive) or equivalent |
| $v_Q = \frac{u(8e-1)}{3}$ | A1 | Both correct (must be positive) or equivalent |
---
## Question 4(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{u(1+4e)}{3} > 0$ | M1 | Working from a correct inequality for their $v_P$ |
| Always true because $e \geq 0$ $\left(\text{or } e > \frac{1}{8}\right)$ | A1 | Correct justification from correct work |
---
## Question 4(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $e = \frac{3}{4} \Rightarrow v_Q = \frac{5u}{3}$ | B1 | Seen or implied |
| Speed of $Q$ after collision $= fv_Q$ | M1 | Impact law for collision with the wall |
| To collide with $P$: $fv_2 > v_1 = \frac{4u}{3}$ | M1 | Correct inequality for second collision |
| $1 \geq f > \frac{4}{5}$ | A1 | Both limits required |
---\begin{enumerate}
\item A particle $P$ of mass $2 m$ is moving in a straight line with speed $u$ on a smooth horizontal plane. The particle $P$ collides directly with a particle $Q$, of mass $m$, which is moving on the plane along the same straight line as $P$ but in the opposite direction to $P$. Immediately before the collision the speed of $Q$ is $3 u$. The coefficient of restitution between $P$ and $Q$ is $e$, where $e > \frac { 1 } { 8 }$\\
(a) Find, in terms of $u$ and $e$,\\
(i) the speed of $P$ immediately after the collision,\\
(ii) the speed of $Q$ immediately after the collision.\\
(b) Show that, for all possible values of $e$, the direction of motion of $P$ is reversed by the collision.
\end{enumerate}
After the collision, $Q$ strikes a smooth fixed vertical wall, which is perpendicular to the direction of motion of $Q$, and rebounds. The coefficient of restitution between $Q$ and the wall is $f$.
Given that $e = \frac { 3 } { 4 }$ and that there is a second collision between $Q$ and $P$,\\
(c) find the range of possible values of $f$.
\hfill \mbox{\textit{Edexcel M2 2018 Q4 [13]}}