| Exam Board | Edexcel |
|---|---|
| Module | FM1 AS (Further Mechanics 1 AS) |
| Year | 2024 |
| Session | June |
| Marks | 11 |
| 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 Further Mechanics 1 collision problem with straightforward application of conservation of momentum and Newton's restitution law. Part (a) uses given information to find one unknown, part (b) applies standard kinetic energy formulas, and part (c) requires determining when a second collision occurs—all routine techniques for FM1 with no novel insight required. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Use of CLM OR NEL | M1 | CLM: Correct no. of terms, condone consistent extra \(g\)'s, sign errors, cancelled \(m\)'s OR NEL: \(e\) on the correct side but condone sign errors |
| \(mu = -m\frac{u}{5}(4e - 1) + 4mv_Q\) OR \(v_Q + \frac{u}{5}(4e - 1) = eu\) | A1 | Correct equation |
| \(v_Q = \frac{u}{5}(e + 1)\) | A1 | Cao. Accept any equivalent two term expression. |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| \(v_P = \pm\frac{fu}{5}(4e - 1)\) | B1 | Seen or implied. |
| \(= \pm\frac{2fu}{5}\) | B1 | Seen or implied. |
| KE Loss \(= \frac{1}{2}m\left(\frac{2u}{5}\right)^2 - \frac{1}{2}m\left(\frac{2fu}{5}\right)^2\) | M1 | Allow negative of this and without \(e\) being substituted. N.B. Allow anything of the form: \(\pm\left(\frac{1}{2}m(v_P)^2 - \frac{1}{2}m(fv_P)^2\right)\), provided that \(v_P\) has come from an attempt to put \(e = \frac{3}{4}\) in the given expression |
| \(= \frac{2mu^2}{25}(1 - f^2)\) | A1 | Cao. Accept any equivalent two term expression, isw |
| (4) |
| Answer | Marks | Guidance |
|---|---|---|
| \(v_Q = \frac{7u}{20}\) | M1 | |
| \(\frac{7u}{20} < \frac{2fu}{5}\) | M1 | Correct inequality for their speeds (which could involve \(e\)), provided it's dimensionally correct. Not available if \(Q\) is moving towards the wall. |
| \(\frac{7}{8} < f\) oe | A1 | |
| \(f < 1\) | B1 | For \(f < 1\) |
| (4) | ||
| (11 marks) | N.B. All the marks are available if they go straight to \(\frac{7}{20} < \frac{2f}{5}\) |
## 4a
| Use of CLM OR NEL | M1 | CLM: Correct no. of terms, condone consistent extra $g$'s, sign errors, cancelled $m$'s OR NEL: $e$ on the correct side but condone sign errors |
| $mu = -m\frac{u}{5}(4e - 1) + 4mv_Q$ OR $v_Q + \frac{u}{5}(4e - 1) = eu$ | A1 | Correct equation |
| $v_Q = \frac{u}{5}(e + 1)$ | A1 | Cao. Accept any equivalent two term expression. |
| | (3) | |
## 4b
| $v_P = \pm\frac{fu}{5}(4e - 1)$ | B1 | Seen or implied. |
| $= \pm\frac{2fu}{5}$ | B1 | Seen or implied. |
| KE Loss $= \frac{1}{2}m\left(\frac{2u}{5}\right)^2 - \frac{1}{2}m\left(\frac{2fu}{5}\right)^2$ | M1 | Allow negative of this and without $e$ being substituted. **N.B. Allow anything of the form:** $\pm\left(\frac{1}{2}m(v_P)^2 - \frac{1}{2}m(fv_P)^2\right)$, provided that $v_P$ has come from an attempt to put $e = \frac{3}{4}$ in the given expression |
| $= \frac{2mu^2}{25}(1 - f^2)$ | A1 | Cao. Accept any equivalent two term expression, isw |
| | (4) | |
## 4c
| $v_Q = \frac{7u}{20}$ | M1 | |
| $\frac{7u}{20} < \frac{2fu}{5}$ | M1 | Correct inequality for their speeds (which could involve $e$), provided it's dimensionally correct. Not available if $Q$ is moving towards the wall. |
| $\frac{7}{8} < f$ oe | A1 | |
| $f < 1$ | B1 | For $f < 1$ |
| | (4) | |
| | (11 marks) | **N.B. All the marks are available if they go straight to** $\frac{7}{20} < \frac{2f}{5}$ |
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{885dd96e-ecaa-4a7f-acb4-f5cf636f491b-10_232_887_246_589}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A particle $P$ of mass $m$ and a particle $Q$ of mass $4 m$ are at rest on a smooth horizontal plane, as shown in Figure 2.
Particle $P$ is projected with speed $u$ along the plane towards $Q$ and the particles collide.\\
The coefficient of restitution between the particles is $e$, where $e > \frac { 1 } { 4 }$
As a result of the collision, the direction of motion of $P$ is reversed and $P$ has speed $\frac { u } { 5 } ( 4 e - 1 )$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $u$ and $e$, the speed of $Q$ after the collision.
After the collision, $P$ goes on to hit a vertical wall which is fixed at right angles to the direction of motion of $P$.
The coefficient of restitution between $P$ and the wall is $f$, where $f > 0$\\
Given that $e = \frac { 3 } { 4 }$
\item find, in terms of $m , u$ and $f$, the kinetic energy lost by $P$ as a result of its impact with the wall. Give your answer in its simplest form.
After its impact with the wall, $P$ goes on to collide with $Q$ again.
\item Find the complete range of possible values of $f$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM1 AS 2024 Q4 [11]}}