| Exam Board | Edexcel |
|---|---|
| Module | FM1 AS (Further Mechanics 1 AS) |
| Year | 2022 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Challenging +1.2 This is a standard Further Mechanics 1 collision problem with multiple parts requiring conservation of momentum, Newton's restitution law, and analysis of a second collision condition. Parts (a)-(c) are routine FM1 bookwork requiring algebraic manipulation of standard formulae. Part (d) requires setting up an inequality for the second collision to occur, which adds modest problem-solving beyond pure recall, but the overall structure follows a familiar FM1 template with no particularly novel insights required. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Use of CLM | M1 | Correct no. of terms, condone sign errors, allow consistently cancelled \(m\)'s or extra \(g\)'s or common factors throughout |
| \(2m \times 2u = -2mw + 3mv\) | A1 | Correct equation; they may have \(w\) instead of \(-w\) |
| Use of NEL | M1 | Correct no. of terms, condone sign errors. M0 if \(e\) on the wrong side of the equation |
| \(2ue = w + v\) | A1 | Correct equation; they may have \(w\) instead of \(-w\) |
| Solve for \(v\) | DM1 | Solve for \(v\), dependent on previous two marks |
| \(v = \dfrac{4u(1+e)}{5}\) | A1* | Correct answer correctly obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Since \(0 \leq e \leq 1\), \(\dfrac{4u(1+0)}{5} \leq v \leq \dfrac{4u(1+1)}{5}\) | M1 | Use of \(0 \leq e \leq 1\) in the given answer; allow use of \(e=0\) and \(e=1\) to obtain min and max expressions. M1A0 for 'verification' |
| i.e. \(\dfrac{4u}{5} \leq v \leq \dfrac{8u}{5}\) | A1* | Correct answer correctly obtained (including use of max and min) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Solve for \(w\) | M1 | Solve for their \(w\) |
| \(w = \dfrac{2u(3e-2)}{5}\) oe \((\text{ms}^{-1})\) or \(\left\ | \dfrac{2u(2-3e)}{5}\right\ | \) oe |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Speed of \(Q\) after hitting the wall \(= \dfrac{1}{6}v\) \((\text{ms}^{-1})\) | M1 | Speed so must see a positive quantity. M0 if \(\frac{1}{6}\) is on the wrong side of the equation |
| For a further collision between \(P\) and \(Q\): \(\dfrac{1}{6}v > w\) | M1 | Correct inequality for their \(w\) (allow even if their \(w\) is dimensionally incorrect) |
| Substitute for \(v\) and \(w\) and solve for \(e\) | M1 | Independent M mark but must have an inequality in \(v\) and \(w\): substitute for \(v\), using given answer, and \(w\) and solve for \(e\) |
| \(e < \dfrac{7}{8}\) | A1 | Correct upper bound for \(e\) |
| \(\dfrac{2}{3} < e < \dfrac{7}{8}\) | A1 | cao |
## Question 4:
### Part 4(a):
**Setup diagram:**
- $2u \rightarrow \quad 0$
- $P(2m) \quad Q(3m)$
- $w \leftarrow \quad \rightarrow v$
| Working/Answer | Mark | Guidance |
|---|---|---|
| Use of CLM | M1 | Correct no. of terms, condone sign errors, allow consistently cancelled $m$'s or extra $g$'s or common factors throughout |
| $2m \times 2u = -2mw + 3mv$ | A1 | Correct equation; they may have $w$ instead of $-w$ |
| Use of NEL | M1 | Correct no. of terms, condone sign errors. M0 if $e$ on the wrong side of the equation |
| $2ue = w + v$ | A1 | Correct equation; they may have $w$ instead of $-w$ |
| Solve for $v$ | DM1 | Solve for $v$, dependent on previous two marks |
| $v = \dfrac{4u(1+e)}{5}$ | A1* | Correct answer correctly obtained |
**(6 marks)**
---
### Part 4(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Since $0 \leq e \leq 1$, $\dfrac{4u(1+0)}{5} \leq v \leq \dfrac{4u(1+1)}{5}$ | M1 | Use of $0 \leq e \leq 1$ in the given answer; allow use of $e=0$ and $e=1$ to obtain min and max expressions. M1A0 for 'verification' |
| i.e. $\dfrac{4u}{5} \leq v \leq \dfrac{8u}{5}$ | A1* | Correct answer correctly obtained (including use of max and min) |
**(2 marks)**
---
### Part 4(c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Solve for $w$ | M1 | Solve for their $w$ |
| $w = \dfrac{2u(3e-2)}{5}$ oe $(\text{ms}^{-1})$ or $\left\|\dfrac{2u(2-3e)}{5}\right\|$ oe | A1 | cao |
**(2 marks)**
---
### Part 4(d):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Speed of $Q$ after hitting the wall $= \dfrac{1}{6}v$ $(\text{ms}^{-1})$ | M1 | Speed so must see a positive quantity. M0 if $\frac{1}{6}$ is on the wrong side of the equation |
| For a further collision between $P$ and $Q$: $\dfrac{1}{6}v > w$ | M1 | Correct inequality for their $w$ (allow even if their $w$ is dimensionally incorrect) |
| Substitute for $v$ and $w$ and solve for $e$ | M1 | Independent M mark but must have an inequality in $v$ and $w$: substitute for $v$, using given answer, and $w$ and solve for $e$ |
| $e < \dfrac{7}{8}$ | A1 | Correct upper bound for $e$ |
| $\dfrac{2}{3} < e < \dfrac{7}{8}$ | A1 | cao |
**(5 marks)**
**Total: 15 marks**\begin{enumerate}
\item A particle $P$ of mass $2 m \mathrm {~kg}$ is moving with speed $2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a smooth horizontal plane. Particle $P$ collides with a particle $Q$ of mass $3 m \mathrm {~kg}$ which is at rest on the plane. The coefficient of restitution between $P$ and $Q$ is $e$. Immediately after the collision the speed of $Q$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$\\
(a) Show that $v = \frac { 4 u ( 1 + e ) } { 5 }$\\
(b) Show that $\frac { 4 u } { 5 } \leqslant v \leqslant \frac { 8 u } { 5 }$
\end{enumerate}
Given that the direction of motion of $P$ is reversed by the collision,\\
(c) find, in terms of $u$ and $e$, the speed of $P$ immediately after the collision.
After the collision, $Q$ hits a wall, that is fixed at right angles to the direction of motion of $Q$, and rebounds.
The coefficient of restitution between $Q$ and the wall is $\frac { 1 } { 6 }$\\
Given that $P$ and $Q$ collide again,\\
(d) find the full range of possible values of $e$.
\hfill \mbox{\textit{Edexcel FM1 AS 2022 Q4 [15]}}