| Exam Board | Edexcel |
|---|---|
| Module | FM1 AS (Further Mechanics 1 AS) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Direct collision with given impulse |
| Difficulty | Standard +0.3 This is a standard FM1 collision question requiring application of impulse-momentum theorem and coefficient of restitution formula. The impulse is given directly, making it more straightforward than typical collision problems where students must derive it. Part (b) is a simple recall of the relationship between e=1 and energy conservation. Slightly easier than average FM1 material. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use of Impulse-momentum principle for \(A\) or \(B\) | M1 | Condone sign errors but M0 if dimensionally incorrect e.g. if \(m\) missing |
| \(A\): \(\frac{9mu}{2} = m(v - {-2u})\) or \(B\): \(\frac{9mu}{2} = 3m(w - {-u})\) | A1 | Correct unsimplified equation |
| Use of Impulse-momentum principle for \(B\) or \(A\) or CLM | M1 | Condone sign errors but M0 if dimensionally incorrect. For CLM, allow consistent missing \(m\)'s or extra \(g\)'s |
| \(\frac{9mu}{2} = 3m(w - {-u})\) or \(\frac{9mu}{2} = m(v - {-2u})\) or \(2mu - 3mu = -mv + 3mw\) | A1 | Correct unsimplified equation |
| \(v = \frac{5u}{2}\) and \(w = \frac{u}{2}\) | A1 | cao for both. Allow one or both negative if correct for their symbols |
| \(e = \dfrac{\frac{5u}{2} + \frac{u}{2}}{2u + u}\) | M1 | Use of NEL. Condone sign errors in numerator but must be terms in \(u\) only AND must be \((2u+u)\) in denominator. M0 if inverted |
| \(e = 1\) | A1cso | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Perfectly elastic (or the coefficient of restitution is 1) so no loss in kinetic energy. \(\frac{1}{2}m(2u)^2 + \frac{1}{2}\times 3mu^2 - \left(\frac{1}{2}m\left(\frac{5u}{2}\right)^2 + \frac{1}{2}\times 3m\left(\frac{u}{2}\right)^2\right) = 0\) | DB1 | Dependent on \(e=1\) correctly obtained in (a). A correct statement e.g. zero, 0 etc and a correct reason. B0 if incorrect extras |
## Question 2(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of Impulse-momentum principle for $A$ or $B$ | M1 | Condone sign errors but M0 if dimensionally incorrect e.g. if $m$ missing |
| $A$: $\frac{9mu}{2} = m(v - {-2u})$ **or** $B$: $\frac{9mu}{2} = 3m(w - {-u})$ | A1 | Correct unsimplified equation |
| Use of Impulse-momentum principle for $B$ or $A$ or CLM | M1 | Condone sign errors but M0 if dimensionally incorrect. For CLM, allow consistent missing $m$'s or extra $g$'s |
| $\frac{9mu}{2} = 3m(w - {-u})$ **or** $\frac{9mu}{2} = m(v - {-2u})$ **or** $2mu - 3mu = -mv + 3mw$ | A1 | Correct unsimplified equation |
| $v = \frac{5u}{2}$ and $w = \frac{u}{2}$ | A1 | cao for both. Allow one or both negative if correct for their symbols |
| $e = \dfrac{\frac{5u}{2} + \frac{u}{2}}{2u + u}$ | M1 | Use of NEL. Condone sign errors in numerator but must be terms in $u$ only AND must be $(2u+u)$ in denominator. M0 if inverted |
| $e = 1$ | A1cso | cso |
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## Question 2(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Perfectly elastic (or the coefficient of restitution is 1) so no loss in kinetic energy. $\frac{1}{2}m(2u)^2 + \frac{1}{2}\times 3mu^2 - \left(\frac{1}{2}m\left(\frac{5u}{2}\right)^2 + \frac{1}{2}\times 3m\left(\frac{u}{2}\right)^2\right) = 0$ | DB1 | Dependent on $e=1$ **correctly** obtained in (a). A correct statement e.g. zero, 0 etc and a correct reason. B0 if incorrect extras |
---\begin{enumerate}
\item Two particles, $A$ and $B$, have masses $m$ and $3 m$ respectively. The particles are moving in opposite directions along the same straight line on a smooth horizontal plane when they collide directly.
\end{enumerate}
Immediately before they collide, $A$ is moving with speed $2 u$ and $B$ is moving with speed $u$.
The direction of motion of each particle is reversed by the collision.\\
In the collision, the magnitude of the impulse exerted on $A$ by $B$ is $\frac { 9 m u } { 2 }$\\
(a) Find the value of the coefficient of restitution between $A$ and $B$.\\
(b) Hence, write down the total loss in kinetic energy due to the collision, giving a reason for your answer.
\hfill \mbox{\textit{Edexcel FM1 AS 2022 Q2 [8]}}