| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, vector velocity form |
| Difficulty | Standard +0.3 This is a standard M4 oblique collision problem requiring conservation of momentum in the direction of impact, Newton's restitution equation, and kinetic energy calculation. The setup is straightforward with given velocities and coefficient of restitution. While it involves multiple steps (resolving velocities, applying collision laws, calculating KE before/after), these are routine procedures for M4 students with no novel insight required. Slightly easier than average due to the clear structure and standard method. |
| Spec | 6.03c Momentum in 2D: vector form6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 3\) and \(b = 2\) | B1 | Both values correct |
| Conservation of linear momentum: \(-4 \times 2 + 3 \times 3 = 2v - 3w\) \((= 1)\) | M1A1 | |
| Restitution: \(v + w = e \times 7\) \((= 3)\) | M1A1 | |
| Solve the simultaneous equations giving \(v = 2\) and \(w = 1\) | DM1, A1 | |
| \(\text{KE lost} = \frac{1}{2} \times 2m \times ((16+9)-(4-9)) + \frac{1}{2} \times 3m \times ((9+4)-(1-4))\) | M1A1 | |
| \(= 24m\) (J) | A1 |
# Question 1:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 3$ and $b = 2$ | B1 | Both values correct |
| Conservation of linear momentum: $-4 \times 2 + 3 \times 3 = 2v - 3w$ $(= 1)$ | M1A1 | |
| Restitution: $v + w = e \times 7$ $(= 3)$ | M1A1 | |
| Solve the simultaneous equations giving $v = 2$ and $w = 1$ | DM1, A1 | |
| $\text{KE lost} = \frac{1}{2} \times 2m \times ((16+9)-(4-9)) + \frac{1}{2} \times 3m \times ((9+4)-(1-4))$ | M1A1 | |
| $= 24m$ (J) | A1 | |
**Total: 10 marks**
---1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{2b891a9c-3abe-4e88-ba94-b6abcb37b4c3-02_794_1022_214_488}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Two smooth uniform spheres $A$ and $B$ have masses $2 m \mathrm {~kg}$ and $3 m \mathrm {~kg}$ respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere $A$ has velocity $( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and sphere $B$ has velocity $( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. When the spheres collide, the line joining their centres is parallel to $\mathbf { j }$, as shown in Figure 1. The coefficient of restitution between the spheres is $\frac { 3 } { 7 }$. Find, in terms of $m$, the total kinetic energy lost in the collision.
\hfill \mbox{\textit{Edexcel M4 2011 Q1 [10]}}