| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2014 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Coalescing particles collision |
| Difficulty | Challenging +1.2 This is a multi-part M3 question combining circular motion with energy conservation and collision mechanics. Part (a) requires energy conservation to find speed at B, then momentum conservation for the collision—both standard techniques. Part (b) applies circular motion formula T = mv²/r + mg. Part (c) requires finding the condition for complete circular motion (tension ≥ 0 at top), involving energy conservation again. While it requires multiple steps and careful bookkeeping across three parts, each individual technique is standard M3 material with no novel insights required. The 'show that' format provides target answers to work towards, reducing difficulty. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.05d Variable speed circles: energy methods |
**Part (a)**
M1 for an energy equation from A to B. Two KE terms and 2 PE terms (or a loss of PE) needed.
$$\frac{1}{2}(2m)v^2 = \frac{1}{2}(2m)u^2 - 2mga(1 - \cos 60°)$$
A1 for correct KE terms (difference either way round)
A1 for a correct loss of PE and all signs correct throughout the equation. Mass can be $m$ or $2m$ for these two A marks, provided consistent.
B1 for a correct conservation of momentum equation
$$2mv = 3mV$$
M1 dep for using the two equations to obtain the speed of the combined particle. Dep on the first M mark and using the C of M equation even if B0 has been given for it.
A1 cso for $V = \sqrt{\frac{u^2 - ag}{3}}$ or $V^2 = \frac{u^2 - ag}{3}$
**Part (b)**
M1 for using NL2 at the bottom, tension, weight and mass $\times$ accel terms required. Accel can be in either form.
$$T - 3mg = 3m \frac{V^2}{a}$$
A1 for a fully correct equation, no need to substitute for the speed.
A1 for substituting the speed (as given in (a)) to obtain a correct expression for the tension in terms of $a$, $g$, $m$ and $u$. Must be simplified.
$$T = 3m\left(\frac{u^2 + 4ag}{3a} - g\right)$$ or equivalent eg $\frac{m}{3a}(12u^2 + 13ag)$
**Part (c)**
M1 for an energy equation from the bottom to the top. Must have a difference of KE terms and a gain of PE.
$$\frac{1}{2}(3m)\left(6 + \frac{10}{12}\right)^2 = \frac{1}{2}(3m)X^2 + 3mg(2a)$$
A1 for a fully correct equation
M1 for NL2 along the radius at the top. Must have a tension, weight and mass $\times$ acceleration (in either form).
$$T + 3mg = 3m\frac{X^2}{a}$$
A1 for a fully correct equation, acceleration in either form.
M1 dep for using $T \geq 0$ at the top to obtain an inequality for the speed at the top and completing to an inequality for $u^2$. Dependent on both previous M marks in (c).
OR: Eliminate $X^2$ between the two equations and then use the inequality $T \geq 0$
A1 cso for $u^2 \geq \frac{41ag}{4}$
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-09_485_442_221_758}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
A particle $P$ of mass $2 m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. Initially the particle is at the point $A$ where $O A = a$ and $O A$ makes an angle $60 ^ { \circ }$ with the downward vertical. The particle is projected downwards from $A$ with speed $u$ in a direction perpendicular to the string, as shown in Figure 3. The point $B$ is vertically below $O$ and $O B = a$. As $P$ passes through $B$ it strikes and adheres to another particle $Q$ of mass $m$ which is at rest at $B$.
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of the combined particle immediately after the impact is
$$\frac { 2 } { 3 } \sqrt { u ^ { 2 } + a g } .$$
\item Find, in terms of $a , g , m$ and $u$, the tension in the string immediately after the impact.
The combined particle moves in a complete circle.
\item Show that $u ^ { 2 } \geqslant \frac { 41 a g } { 4 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2014 Q5 [15]}}