| Exam Board | Edexcel |
|---|---|
| Module | FM1 AS (Further Mechanics 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Multiple wall bounces or returns |
| Difficulty | Standard +0.8 This FM1 question requires applying coefficient of restitution across multiple collisions, deriving an expression for cumulative kinetic energy loss (involving algebraic manipulation with powers of e), then optimizing by differentiation. The multi-step nature, algebraic complexity with e terms, and the optimization component make it moderately challenging for Further Maths AS level, though the underlying physics principles are standard. |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.02e Calculate KE and PE: using formulae6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Speeds after 1st and 2nd impacts: \(eu\) and \(e^2u\) | B1 | Need both for the mark |
| KE Loss, \(K = \frac{1}{2}emu^2 - \frac{1}{2}em(e^2u)^2\) (difference in KE's) | M1 | Allow terms reversed |
| \(\frac{1}{2}mu^2(e - e^5)\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate wrt \(e\) | M1 | Clear attempt to differentiate their KE loss, in terms of \(e\), wrt \(e\), with powers decreasing by 1 |
| \(\frac{dK}{de} = \frac{1}{2}mu^2(1 - 5e^4)\) | A1 | Correct derivative. If working from \(\frac{1}{2}mu^2(1-e^2)\) allow M1A0 for a correct argument leading to \(e=0\) |
| Equate to zero and solve for \(e\) | M1 | Clear attempt to equate to zero |
| \(e^4 = \frac{1}{5} \Rightarrow e = 0.67\) or better | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Particle continues to bounce off each wall (indefinitely) | B1 | Any clear equivalent statement |
| Speed of particle decreases | B1 | Any clear equivalent statement. Allow speed tends to 0 |
## Question 2:
**Part (a):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Speeds after 1st and 2nd impacts: $eu$ and $e^2u$ | B1 | Need both for the mark |
| KE Loss, $K = \frac{1}{2}emu^2 - \frac{1}{2}em(e^2u)^2$ (difference in KE's) | M1 | Allow terms reversed |
| $\frac{1}{2}mu^2(e - e^5)$ | A1 | cao |
**Part (b):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate wrt $e$ | M1 | Clear attempt to differentiate their KE loss, in terms of $e$, wrt $e$, with powers decreasing by 1 |
| $\frac{dK}{de} = \frac{1}{2}mu^2(1 - 5e^4)$ | A1 | Correct derivative. If working from $\frac{1}{2}mu^2(1-e^2)$ allow M1A0 for a correct argument leading to $e=0$ |
| Equate to zero and solve for $e$ | M1 | Clear attempt to equate to zero |
| $e^4 = \frac{1}{5} \Rightarrow e = 0.67$ or better | A1 | cao |
**Part (c):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Particle continues to bounce off each wall (indefinitely) | B1 | Any clear equivalent statement |
| Speed of particle decreases | B1 | Any clear equivalent statement. Allow speed tends to 0 |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{05f6f654-05e5-41d5-a6e4-11cd91a6df83-06_458_278_248_986}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A particle of mass em is at rest on a smooth horizontal plane between two smooth fixed parallel vertical walls, as shown in the plan view in Figure 2. The particle is projected along the plane with speed $u$ towards one of the walls and strikes the wall at right angles. The coefficient of restitution between the particle and each wall is $e$ and air resistance is modelled as being negligible.
Using the model,
\begin{enumerate}[label=(\alph*)]
\item find, in terms of $m , u$ and $e$, an expression for the total loss in the kinetic energy of the particle as a result of the first two impacts.
Given that $e$ can vary such that $0 < e < 1$ and using the model,
\item find the value of $e$ for which the total loss in the kinetic energy of the particle as a result of the first two impacts is a maximum,
\item describe the subsequent motion of the particle.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM1 AS 2021 Q2 [9]}}