| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2020 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Direct collision with energy loss |
| Difficulty | Standard +0.3 This is a standard M2 collision problem requiring conservation of momentum, coefficient of restitution formula, and energy loss calculation. While it involves multiple steps and algebraic manipulation with the given energy loss condition, it follows a well-established procedure that students practice extensively. The problem is slightly easier than average because the direction reversals provide strong constraints that simplify the algebra. |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.02i Conservation of energy: mechanical energy principle6.03b Conservation of momentum: 1D two particles6.03e Impulse: by a force6.03f Impulse-momentum: relation6.03j Perfectly elastic/inelastic: collisions |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{4m}{2}(4u^2 - v^2) + \frac{3m}{2}(9u^2 - w^2) = \frac{473}{24}mu^2\) | M1 | Need all terms. Dimensionally correct. Accept \(\pm\) |
| \((48v^2 + 36w^2 = 43u^2)\) | A1 | Correct unsimplified equation in \(v\), \(w\) or their \(v\), \(w\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(8mu - 9mu = -4mv + 3mw\) | M1 | Need all terms. Dimensionally correct. Condone sign errors. |
| \((u = 4v - 3w)\) | A1 | Correct unsimplified equation with their correct signs |
| Answer | Marks | Guidance |
|---|---|---|
| \(w + v = 5eu\) | M1 | Must be used the right way round |
| A1 | Or equivalent in their \(w\), \(v\). Signs for \(v\), \(w\) consistent with CLM eqn |
| Answer | Marks | Guidance |
|---|---|---|
| \(48v^2 + 36\left(\frac{4v-u}{3}\right)^2 = 43u^2\) or \(48\left(\frac{u+3w}{4}\right)^2 + 36w^2 = 43u^2\) or \(\frac{48}{49}(1+15e)^2 + \frac{36}{49}(20e-1)^2 = 43\) | DM1 | Form equation for \(v\) or \(w\) or \(e\). Dependent on M marks scored for equations used. |
| Answer | Marks | Guidance |
|---|---|---|
| \(112v^2 - 32uv - 39u^2 = 0 = (4v-3u)(28v+13u)\) or \(63w^2 + 18uw - 40u^2 = 0 = (21w+20u)(3w-2u)\) or \(25200e^2 = 2023\) | DM1 | Solve for \(v\) or \(w\) or \(e\). Dependent on the preceding M |
| Answer | Marks | Guidance |
|---|---|---|
| \(v = \frac{3u}{4}\), \(w = \frac{2u}{3}\) | A1 | \(v\) or \(w\) correct |
| \(\frac{3u}{4} + \frac{2u}{3} = 5eu\), \(e = \frac{17}{60}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Use of \(I = m(v - u)\) | DM1 | Must be attempting to subtract corresponding values for \(u\) and \(v\). Dependent on the first 4 M marks. |
| \(4m\left(2u + \frac{3u}{4}\right) = 11mu\) | A1 (12) | Or \(3m\left(3u + \frac{2u}{3}\right)\) from correct solution only |
# Question 8:
## Change in KE
| $\frac{4m}{2}(4u^2 - v^2) + \frac{3m}{2}(9u^2 - w^2) = \frac{473}{24}mu^2$ | M1 | Need all terms. Dimensionally correct. Accept $\pm$ |
|---|---|---|
| $(48v^2 + 36w^2 = 43u^2)$ | A1 | Correct unsimplified equation in $v$, $w$ or their $v$, $w$ |
## Conservation of Linear Momentum (CLM)
| $8mu - 9mu = -4mv + 3mw$ | M1 | Need all terms. Dimensionally correct. Condone sign errors. |
|---|---|---|
| $(u = 4v - 3w)$ | A1 | Correct unsimplified equation with their correct signs |
## Impact Law
| $w + v = 5eu$ | M1 | Must be used the right way round |
|---|---|---|
| | A1 | Or equivalent in their $w$, $v$. Signs for $v$, $w$ consistent with CLM eqn |
## Forming equation
| $48v^2 + 36\left(\frac{4v-u}{3}\right)^2 = 43u^2$ or $48\left(\frac{u+3w}{4}\right)^2 + 36w^2 = 43u^2$ or $\frac{48}{49}(1+15e)^2 + \frac{36}{49}(20e-1)^2 = 43$ | DM1 | Form equation for $v$ or $w$ or $e$. Dependent on M marks scored for equations used. |
## Solving
| $112v^2 - 32uv - 39u^2 = 0 = (4v-3u)(28v+13u)$ or $63w^2 + 18uw - 40u^2 = 0 = (21w+20u)(3w-2u)$ or $25200e^2 = 2023$ | DM1 | Solve for $v$ or $w$ or $e$. Dependent on the preceding M |
## Final answers
| $v = \frac{3u}{4}$, $w = \frac{2u}{3}$ | A1 | $v$ or $w$ correct |
|---|---|---|
| $\frac{3u}{4} + \frac{2u}{3} = 5eu$, $e = \frac{17}{60}$ | A1 | |
## Impulse
| Use of $I = m(v - u)$ | DM1 | Must be attempting to subtract corresponding values for $u$ and $v$. Dependent on the first 4 M marks. |
|---|---|---|
| $4m\left(2u + \frac{3u}{4}\right) = 11mu$ | A1 (12) | Or $3m\left(3u + \frac{2u}{3}\right)$ from correct solution only |
**Total: [12]**\begin{enumerate}
\item A particle $A$ has mass $4 m$ and a particle $B$ has mass $3 m$. The particles are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide directly.
\end{enumerate}
Immediately before the collision the speed of $A$ is $2 u$ and the speed of $B$ is $3 u$.\\
The direction of motion of each particle is reversed by the collision.\\
The total kinetic energy lost in the collision is $\frac { 473 } { 24 } m u ^ { 2 }$\\
Find\\
(i) the coefficient of restitution between $A$ and $B$,\\
(ii) the magnitude of the impulse received by $A$ in the collision.\\
\hfill \mbox{\textit{Edexcel M2 2020 Q8 [12]}}