| Exam Board | Edexcel |
|---|---|
| Module | FM1 (Further Mechanics 1) |
| Session | Specimen |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Range of coefficient of restitution |
| Difficulty | Standard +0.8 This FM1 question requires applying conservation of momentum and the coefficient of restitution formula simultaneously, then using the constraint that Q's direction reverses to establish inequalities for e. Part (b) involves algebraic manipulation of kinetic energy expressions, and part (c) tests conceptual understanding of how e relates to energy loss. The multi-step reasoning and inequality work elevate this above standard mechanics problems. |
| Spec | 6.03a Linear momentum: p = mv6.03b 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 |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Complete overall strategy to find \(v\) | M1 | Complete strategy to form sufficient equations in \(v\) and \(w\) and solve for \(v\) |
| Use of CLM | M1 | Use CLM to form equation in \(v\) and \(w\); needs all 4 terms & dimensionally correct |
| \(2m \times 2u - 5m \times u = 5m \times v - 2m \times w\), \((-u = 5v - 2w)\) | A1 | Correct unsimplified equation |
| Use of Impact law: \(v + w = e(2u + u)\) | M1 A1 | Use NEL as a model to form a second equation in \(v\) and \(w\); must be used the right way round |
| Solve for \(v\): \(-u = 5v - 2w\), \(6eu = 2v + 2w\) | ||
| \(7v = u(6e-1)\), \(\left(v = \frac{u}{7}(6e-1)\right)\) | A1 | for \(v\) or \(7v\) correct |
| Direction of \(Q\) reversed: \(v > 0\) | M1 | Use the model to form a correct inequality for their \(v\) |
| \(\Rightarrow 1 \geq e > \frac{1}{6}\) | A1 | Both limits required |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(e = \frac{1}{3} \Rightarrow v = \frac{u}{7}\), \(w = \frac{6u}{7}\) | B1 | Or equivalent statements |
| Equation for KE lost | M1 | Terms of correct structure combined correctly |
| \(\frac{1}{2} \times 2m\left(4u^2 - \frac{36u^2}{49}\right) + \frac{1}{2} \times 5m\left(u^2 - \frac{u^2}{49}\right)\) | A1 A1 | Fully correct unsimplified A1A1; one error on unsimplified expression A1A0 |
| \(\frac{1}{2}mu^2\left(8 - \frac{72}{49} + 5 - \frac{5}{49}\right) = \frac{40mu^2}{7}\) | A1* | cso, plus a statement that the required result has been achieved |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Increase \(e \Rightarrow\) more elastic \(\Rightarrow\) less energy lost | B1 | "less energy lost" or equivalent |
## Question 8:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Complete overall strategy to find $v$ | M1 | Complete strategy to form sufficient equations in $v$ and $w$ and solve for $v$ |
| Use of CLM | M1 | Use CLM to form equation in $v$ and $w$; needs all 4 terms & dimensionally correct |
| $2m \times 2u - 5m \times u = 5m \times v - 2m \times w$, $(-u = 5v - 2w)$ | A1 | Correct unsimplified equation |
| Use of Impact law: $v + w = e(2u + u)$ | M1 A1 | Use NEL as a model to form a second equation in $v$ and $w$; must be used the right way round |
| Solve for $v$: $-u = 5v - 2w$, $6eu = 2v + 2w$ | | |
| $7v = u(6e-1)$, $\left(v = \frac{u}{7}(6e-1)\right)$ | A1 | for $v$ or $7v$ correct |
| Direction of $Q$ reversed: $v > 0$ | M1 | Use the model to form a correct inequality for their $v$ |
| $\Rightarrow 1 \geq e > \frac{1}{6}$ | A1 | Both limits required |
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $e = \frac{1}{3} \Rightarrow v = \frac{u}{7}$, $w = \frac{6u}{7}$ | B1 | Or equivalent statements |
| Equation for KE lost | M1 | Terms of correct structure combined correctly |
| $\frac{1}{2} \times 2m\left(4u^2 - \frac{36u^2}{49}\right) + \frac{1}{2} \times 5m\left(u^2 - \frac{u^2}{49}\right)$ | A1 A1 | Fully correct unsimplified A1A1; one error on unsimplified expression A1A0 |
| $\frac{1}{2}mu^2\left(8 - \frac{72}{49} + 5 - \frac{5}{49}\right) = \frac{40mu^2}{7}$ | A1* | cso, plus a statement that the required result has been achieved |
### Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Increase $e \Rightarrow$ more elastic $\Rightarrow$ less energy lost | B1 | "less energy lost" or equivalent |\begin{enumerate}
\item A particle $P$ of mass $2 m$ and a particle $Q$ of mass $5 m$ are moving along the same straight line on a smooth horizontal plane.
\end{enumerate}
They are moving in opposite directions towards each other and collide directly.\\
Immediately before the collision the speed of $P$ is $2 u$ and the speed of $Q$ is $u$.\\
The direction of motion of $Q$ is reversed by the collision.\\
The coefficient of restitution between $P$ and $Q$ is $e$.\\
(a) Find the range of possible values of $e$.
Given that $e = \frac { 1 } { 3 }$\\
(b) show that the kinetic energy lost in the collision is $\frac { 40 m u ^ { 2 } } { 7 }$.\\
(c) Without doing any further calculation, state how the amount of kinetic energy lost in the collision would change if $e > \frac { 1 } { 3 }$
\hfill \mbox{\textit{Edexcel FM1 Q8 [14]}}