| Exam Board | Edexcel |
|---|---|
| Module | FM1 (Further Mechanics 1) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.8 This is a Further Mechanics 1 question requiring conservation of momentum and Newton's restitution law applied twice (particle collision then wall impact), followed by inequality analysis to determine conditions for a second collision. Part (a) is a standard 'show that' requiring careful algebraic manipulation. Part (b) requires deeper reasoning about relative velocities and direction changes, making this moderately challenging for FM1 but still following established collision mechanics patterns. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| CLM: \(2m \times 3u - m \times 2u = 2mv_P + mv_Q \quad (4u = 2v_P + v_Q)\) | M1, A1 | CLM used. Dimensionally correct, mass × velocity. All terms required. Condone sign errors. Condone consistent \(g\)'s or cancelled \(m\)'s |
| Impact Law: \(5ue = -v_P + v_Q\) | M1, A1 | NEL used correctly with \(e\) on correct side. Condone sign errors, must have correct number of terms. Direction of \(v_Q\) and \(v_P\) must be consistent with CLM equation |
| Attempts to solve for \(v_Q\) | dM1 | At least one line of working should be seen |
| \(v_Q = \frac{(4+10e)u}{3}\) | A1* | Also accept: \(\frac{1}{3}(4+10e)u\), \(\frac{u(4+10e)}{3}\), \(\frac{u}{3}(4+10e)\). Do not accept \(\frac{(10e+4)u}{3}\) — this is A0* |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(v_P = \frac{(4-5e)u}{3}\) oe | M1 | Attempt to solve for \(v_P\). If \(v_P\) found in (a) it must be used here. If \(P\) assumed to reverse: \(v_P = \frac{(5e-4)u}{3}\) oe |
| Correct rebound speed or velocity of \(Q\) seen \(\pm\frac{f(4+10e)u}{3}\) | B1 | Correct expression for speed/velocity of \(Q\) after rebound. May appear on diagram |
| States correct inequality e.g. \(\frac{f(4+10e)u}{3} > -\frac{(4-5e)u}{3}\) | M1 | Correct unsimplified inequality. Accepts cancelled \(u\)'s and/or 3's |
| \((1\geq) f > \frac{5e-4}{4+10e}\) | A1 | Correct inequality, do not ISW. Allow \(1 \geq f\) to be omitted but do not allow strict inequality \(1 > f\) |
# Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| CLM: $2m \times 3u - m \times 2u = 2mv_P + mv_Q \quad (4u = 2v_P + v_Q)$ | M1, A1 | CLM used. Dimensionally correct, mass × velocity. All terms required. Condone sign errors. Condone consistent $g$'s or cancelled $m$'s |
| Impact Law: $5ue = -v_P + v_Q$ | M1, A1 | NEL used correctly with $e$ on correct side. Condone sign errors, must have correct number of terms. Direction of $v_Q$ and $v_P$ must be consistent with CLM equation |
| Attempts to solve for $v_Q$ | dM1 | At least one line of working should be seen |
| $v_Q = \frac{(4+10e)u}{3}$ | A1* | Also accept: $\frac{1}{3}(4+10e)u$, $\frac{u(4+10e)}{3}$, $\frac{u}{3}(4+10e)$. Do not accept $\frac{(10e+4)u}{3}$ — this is A0* |
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# Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $v_P = \frac{(4-5e)u}{3}$ oe | M1 | Attempt to solve for $v_P$. If $v_P$ found in (a) it must be used here. If $P$ assumed to reverse: $v_P = \frac{(5e-4)u}{3}$ oe |
| Correct rebound speed or velocity of $Q$ seen $\pm\frac{f(4+10e)u}{3}$ | B1 | Correct expression for speed/velocity of $Q$ after rebound. May appear on diagram |
| States correct inequality e.g. $\frac{f(4+10e)u}{3} > -\frac{(4-5e)u}{3}$ | M1 | Correct unsimplified inequality. Accepts cancelled $u$'s and/or 3's |
| $(1\geq) f > \frac{5e-4}{4+10e}$ | A1 | Correct inequality, do not ISW. Allow $1 \geq f$ to be omitted but do not allow strict inequality $1 > f$ |
---\begin{enumerate}
\item A particle $P$ of mass $2 m$ is moving in a straight line with speed $3 u$ on a smooth horizontal plane. It collides directly with a particle $Q$ of mass $m$ that is moving on the plane with speed $2 u$ in the opposite direction to $P$.\\
The coefficient of restitution between $P$ and $Q$ is $e$, where $e > \frac { 4 } { 5 }$\\
(a) Show that the speed of $Q$ immediately after the collision is $\frac { ( 4 + 10 e ) u } { 3 }$
\end{enumerate}
After the collision $Q$ hits a smooth fixed vertical wall that is perpendicular to the direction of motion of $Q$. The coefficient of restitution between $Q$ and the wall is $f$.\\
(b) Find, in terms of $\boldsymbol { e }$, the set of values of $f$ for which there will be a second collision between $P$ and $Q$.
\hfill \mbox{\textit{Edexcel FM1 2023 Q3 [10]}}