| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Three-particle sequential collisions |
| Difficulty | Challenging +1.2 This is a sequential collision problem requiring systematic application of conservation of momentum and Newton's restitution law across two collisions, with inequality constraints from direction reversals. While it involves multiple steps and careful algebraic manipulation, the techniques are standard M2 material with no novel insights required—moderately above average difficulty for A-level. |
| Spec | 6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(mu = -mv + 3mw\) | M1 | CLM. Allow \(v\) in either direction. Needs all 3 terms. Condone sign errors. |
| \(u = -v + 3w\) | A1 | \(v\) in either direction. Ignore diagram if equations "correct" but inconsistent with diagram. |
| \(eu = w + v\) | M1 | Impact law. Must be the right way round, but condone sign errors. |
| Correct equation, signs consistent with CLM equation | A1 | |
| \(w = \dfrac{u}{4}(1+e)\) | DM1 | Solve for \(v\) or \(w\). |
| One correct | A1 | |
| \(v = -w + eu = \dfrac{u}{4}(3e-1)\) | A1 | Both correct. \(1 - 3e \rightarrow\) A0 for \(v\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3mw = 4mX - 3mY\) | M1 | CLM for their \(w\). |
| Correct unsimplified (their \(w\)) | A1ft | |
| \(2ew = X + Y\) | B1ft | Impact law. Must be the right way up. Their \(w\). |
| \(7Y = W(8e-3)\) | DM1 | Solve for \((7)Y\) |
| Or \(2ue(1+e) - \dfrac{3u}{4}(1+e) = 7Y\) | ||
| \(e > \dfrac{3}{8}\) | A1 | |
| \(Y > 0 \rightarrow \dfrac{3}{8} < e \leq \dfrac{1}{2}\) | A1 | NB No longer ft. Condone \(<\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{u}{28}(1+e)(8e-3) > \dfrac{u}{4}(3e-1)\) | M1 | For a second collision their \(Y >\) their \(v\) |
| \(2e^2 - 4e + 1 > 0\) | ||
| \(e = \dfrac{4 \pm \sqrt{16-8}}{4} = 1.707,\ 0.293\) | DM1 | Obtain the critical values |
| \(2e^2 - 4e + 1 < 0\) for \(\dfrac{3}{8} < e \leq \dfrac{1}{2}\), so no second collision. | A1 | Compare \(0.293\) (o.e.) with \(\dfrac{3}{8}\) to reach correct conclusion for correct reason. |
## Question 7:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $mu = -mv + 3mw$ | M1 | CLM. Allow $v$ in either direction. Needs all 3 terms. Condone sign errors. |
| $u = -v + 3w$ | A1 | $v$ in either direction. Ignore diagram if equations "correct" but inconsistent with diagram. |
| $eu = w + v$ | M1 | Impact law. Must be the right way round, but condone sign errors. |
| Correct equation, signs consistent with CLM equation | A1 | |
| $w = \dfrac{u}{4}(1+e)$ | DM1 | Solve for $v$ or $w$. |
| One correct | A1 | |
| $v = -w + eu = \dfrac{u}{4}(3e-1)$ | A1 | Both correct. $1 - 3e \rightarrow$ A0 for $v$ |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3mw = 4mX - 3mY$ | M1 | CLM for their $w$. |
| Correct unsimplified (their $w$) | A1ft | |
| $2ew = X + Y$ | B1ft | Impact law. Must be the right way up. Their $w$. |
| $7Y = W(8e-3)$ | DM1 | Solve for $(7)Y$ |
| Or $2ue(1+e) - \dfrac{3u}{4}(1+e) = 7Y$ | | |
| $e > \dfrac{3}{8}$ | A1 | |
| $Y > 0 \rightarrow \dfrac{3}{8} < e \leq \dfrac{1}{2}$ | A1 | NB No longer ft. Condone $<$. |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{u}{28}(1+e)(8e-3) > \dfrac{u}{4}(3e-1)$ | M1 | For a second collision their $Y >$ their $v$ |
| $2e^2 - 4e + 1 > 0$ | | |
| $e = \dfrac{4 \pm \sqrt{16-8}}{4} = 1.707,\ 0.293$ | DM1 | Obtain the critical values |
| $2e^2 - 4e + 1 < 0$ for $\dfrac{3}{8} < e \leq \dfrac{1}{2}$, so no second collision. | A1 | Compare $0.293$ (o.e.) with $\dfrac{3}{8}$ to reach correct conclusion for correct reason. |7. A particle $A$ of mass $m$ is moving with speed $u$ on a smooth horizontal floor when it collides directly with another particle $B$, of mass $3 m$, which is at rest on the floor. The coefficient of restitution between the particles is $e$. The direction of motion of $A$ is reversed by the collision.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $e$ and $u$,
\begin{enumerate}[label=(\roman*)]
\item the speed of $A$ immediately after the collision,
\item the speed of $B$ immediately after the collision.
After being struck by $A$ the particle $B$ collides directly with another particle $C$, of mass $4 m$, which is at rest on the floor. The coefficient of restitution between $B$ and $C$ is $2 e$. Given that the direction of motion of $B$ is reversed by this collision,
\end{enumerate}\item find the range of possible values of $e$,
\item determine whether there will be a second collision between $A$ and $B$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2013 Q7 [16]}}