| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2017 |
| Session | October |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.8 This is a challenging M2 collision problem requiring conservation of momentum and restitution equations for two separate collisions, followed by algebraic manipulation to derive inequality bounds on e. The multi-stage nature (collision → wall → second collision condition) and the need to establish precise inequalities from physical constraints (B's direction reverses, second collision occurs) elevates this significantly above routine collision exercises. |
| 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2mu = -2mv_B + 3mv_A\) | M1 | CLM. Need all 3 terms. Dimensionally correct. |
| A1 | Correct unsimplified | |
| \(eu = v_A + v_B\) | M1 | Impact law. Used the right way round. Condone sign error |
| A1 | Correct with signs consistent with CLM equation | |
| Solve for \(v_A\) or \(v_B\) | dM1 | For finding either. Dependent on both preceding M marks |
| \(v_A = \dfrac{2u}{5}(1+e)\) | A1 | (i) |
| \(v_B = \dfrac{u}{5}(3e-2)\) | A1 | (ii) They need modulus signs if they have \(2-3e\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Dir of motion of \(B\) reversed \(\Rightarrow v_B > 0\ \Rightarrow e > \dfrac{2}{3}\) | B1 | |
| Impact between \(A\) and the wall: \(\frac{1}{7}\times\frac{2u}{5}(1+e) = V_A\) | B1ft | Follow their \(v_A\) |
| For a second impact between \(A\) and \(B\), \(V_A > v_B\): \(\quad\frac{1}{7}\times\frac{2u}{5}(1+e) > \frac{u}{5}(3e-2)\) | M1 | Inequality must be the right way round. |
| \(2+2e > 21e-14\) | ||
| \(19e < 16\quad e < \dfrac{16}{19}\) | A1 | |
| \(\therefore\ \dfrac{2}{3} < e < \dfrac{16}{19}\) Given Answer | A1 | With sufficient correct working to justify given answer. |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2mu = -2mv_B + 3mv_A$ | M1 | CLM. Need all 3 terms. Dimensionally correct. |
| | A1 | Correct unsimplified |
| $eu = v_A + v_B$ | M1 | Impact law. Used the right way round. Condone sign error |
| | A1 | Correct with signs consistent with CLM equation |
| Solve for $v_A$ or $v_B$ | dM1 | For finding either. Dependent on both preceding M marks |
| $v_A = \dfrac{2u}{5}(1+e)$ | A1 | (i) |
| $v_B = \dfrac{u}{5}(3e-2)$ | A1 | (ii) They need modulus signs if they have $2-3e$ |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Dir of motion of $B$ reversed $\Rightarrow v_B > 0\ \Rightarrow e > \dfrac{2}{3}$ | B1 | |
| Impact between $A$ and the wall: $\frac{1}{7}\times\frac{2u}{5}(1+e) = V_A$ | B1ft | Follow their $v_A$ |
| For a second impact between $A$ and $B$, $V_A > v_B$: $\quad\frac{1}{7}\times\frac{2u}{5}(1+e) > \frac{u}{5}(3e-2)$ | M1 | Inequality must be the right way round. |
| $2+2e > 21e-14$ | | |
| $19e < 16\quad e < \dfrac{16}{19}$ | A1 | |
| $\therefore\ \dfrac{2}{3} < e < \dfrac{16}{19}$ **Given Answer** | A1 | With sufficient correct working to justify given answer. |8. A particle $A$ of mass $3 m$ lies at rest on a smooth horizontal floor. A particle $B$ of mass $2 m$ is moving in a straight line on the floor with speed $u$ when it collides directly with $A$. The coefficient of restitution between $A$ and $B$ is $e$. As a result of the collision the direction of motion of $B$ is reversed.
\begin{enumerate}[label=(\alph*)]
\item Find an expression, in terms of $u$ and $e$, for
\begin{enumerate}[label=(\roman*)]
\item the speed of $A$ immediately after the collision,
\item the speed of $B$ immediately after the collision.
The particle $A$ subsequently strikes a smooth vertical wall. The wall is perpendicular to the direction of motion of $A$. The coefficient of restitution between $A$ and the wall is $\frac { 1 } { 7 }$ There is a second collision between $A$ and $B$.
\end{enumerate}\item Show that $\frac { 2 } { 3 } < e < \frac { 16 } { 19 }$\\
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\hfill \mbox{\textit{Edexcel M2 2017 Q8 [12]}}