| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2002 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.8 This M2 collision problem requires systematic application of conservation of momentum and Newton's restitution law, followed by inequality analysis for a second collision condition. While the individual principles are standard, the multi-stage collision scenario (ball-ball, then ball-wall, then determining conditions for a second ball-ball collision) and the algebraic manipulation to find the range of e values requires careful reasoning and is more demanding than typical M2 questions. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| CoM: \(mu = -mv_1 + 3mv_2\) \(\Rightarrow\) \(u = -v_1 + 3v_2\) | ||
| NEL: \(e u = v_2 + v_1\) | ||
| Solving: \(v_1 = \frac{1}{4}(3c - 1)u\) | ||
| \(v_2 = \frac{1}{4}(1 + e)u\) | ||
| Speed of \(B\) after hitting wall \(= \pm \frac{3}{16}(1 + e)u\) \((v_2^*)\) | ||
| For second collision: \(v_2^* > v_1\); \(\frac{3}{16}(1 + e)u > \frac{1}{4}(3c - 1)u\) | ||
| Solving, \(e < \frac{7}{9}\) | ||
| Finding lower bound using \(v_1 > 0\); \(e > \frac{1}{3}\) | ||
| Complete range: \(\frac{1}{3} < e < \frac{7}{9}\) |
## Part (a)
| CoM: $mu = -mv_1 + 3mv_2$ $\Rightarrow$ $u = -v_1 + 3v_2$ | | | | M1 A1 |
| NEL: $e u = v_2 + v_1$ | | | | M1 A1 |
| Solving: $v_1 = \frac{1}{4}(3c - 1)u$ | | | | M1 A1 |
| | $v_2 = \frac{1}{4}(1 + e)u$ | | | | A1 (7) |
| Speed of $B$ after hitting wall $= \pm \frac{3}{16}(1 + e)u$ $(v_2^*)$ | | | | B1 ft |
| For second collision: $v_2^* > v_1$; $\frac{3}{16}(1 + e)u > \frac{1}{4}(3c - 1)u$ | | | | M1 |
| Solving, $e < \frac{7}{9}$ | | | | M1 A1 |
| Finding lower bound using $v_1 > 0$; $e > \frac{1}{3}$ | | | | M1 |
| Complete range: $\frac{1}{3} < e < \frac{7}{9}$ | | | | A1 (cso) (6) |
**Total: 13 marks**
---A small smooth ball $A$ of mass $m$ is moving on a horizontal table with speed $u$ when it collides directly with another small smooth ball $B$ of mass $3m$ which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is $e$. The direction of motion of $A$ is reversed as a result of the collision.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $e$ and $u$, the speeds of $A$ and $B$ immediately after the collision. [7]
\end{enumerate}
In the subsequent motion $B$ strikes a vertical wall, which is perpendicular to the direction of motion of $B$, and rebounds. The coefficient of restitution between $B$ and the wall is $\frac{1}{3}$.
Given that there is a second collision between $A$ and $B$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the range of values of $e$ for which the motion described is possible. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2002 Q6 [13]}}