| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2004 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Particle brought to rest by collision |
| Difficulty | Moderate -0.3 This is a standard M2 collision problem requiring routine application of conservation of momentum and Newton's restitution law. Parts (a)-(b) involve straightforward algebraic manipulation of two equations with two unknowns. Parts (c)-(d) add a kinetic energy condition but remain mechanical. The 'show that' format provides target answers, reducing problem-solving demand. Slightly easier than average due to its highly structured, textbook nature. |
| Spec | 6.03a Linear momentum: p = mv6.03b 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 |
|---|---|---|
| (a) \(u \to \to 0\) | B1 | |
| \(m \quad 3m\) | ||
| \(v_1 \to \quad v_2 \to\) | ||
| CLM: \(mu = mv_1 + 3 mv_2\) | M1 A1 | |
| NIL: \(eu = -v_1 + v_2\) | ||
| solving, | dep. M1 | |
| \(v_2 = \frac{u}{4}(1 + e)\) * | A1 | (5) |
| (b) Solving for \(v_1\): \(\left | \frac{u}{4}(1 - 3e)\right | \) |
| (c) \(\frac{1}{2} m \frac{u^2}{16} (1 - 3e)^2 + \frac{1}{2} 3m \frac{u^2}{16} (1 + e)^2 = \frac{1}{6} mu^2\) | M1 A1 f.t. A1 | |
| \(e^2 = \frac{1}{9}\) | dep. M1 A1 | |
| \(e = \frac{1}{3}\) | A1 | (6) |
| (d) \(v_1 = \frac{u}{4}(1 - 3 \times \frac{1}{3}) = 0 \Rightarrow\) at rest | A1 c.s.o. | (1) |
| (14 marks) |
**(a)** $u \to \to 0$ | B1 |
$m \quad 3m$ | |
$v_1 \to \quad v_2 \to$ | |
CLM: $mu = mv_1 + 3 mv_2$ | M1 A1 |
NIL: $eu = -v_1 + v_2$ | |
solving, | dep. M1 |
$v_2 = \frac{u}{4}(1 + e)$ * | A1 | (5)
**(b)** Solving for $v_1$: $\left|\frac{u}{4}(1 - 3e)\right|$ | M1 A1 | (2)
**(c)** $\frac{1}{2} m \frac{u^2}{16} (1 - 3e)^2 + \frac{1}{2} 3m \frac{u^2}{16} (1 + e)^2 = \frac{1}{6} mu^2$ | M1 A1 f.t. A1 |
$e^2 = \frac{1}{9}$ | dep. M1 A1 |
$e = \frac{1}{3}$ | A1 | (6)
**(d)** $v_1 = \frac{u}{4}(1 - 3 \times \frac{1}{3}) = 0 \Rightarrow$ at rest | A1 c.s.o. | (1)
| | | **(14 marks)** |
---6. A smooth sphere $A$ of mass $m$ is moving with speed $u$ on a smooth horizontal table when it collides directly with another smooth sphere $B$ of mass $3 m$, which is at rest on the table. The coefficient of restitution between $A$ and $B$ is $e$. The spheres have the same radius and are modelled as particles.
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $B$ immediately after the collision is $\frac { 1 } { 4 } ( 1 + e ) u$.
\item Find the speed of $A$ immediately after the collision.
Immediately after the collision the total kinetic energy of the spheres is $\frac { 1 } { 6 } m u ^ { 2 }$.
\item Find the value of $e$.
\item Hence show that $A$ is at rest after the collision.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2004 Q6 [14]}}