| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2004 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Two-sphere oblique collision |
| Difficulty | Standard +0.8 This M4 oblique collision question requires resolving velocities into components along/perpendicular to the line of centres, applying conservation of momentum and Newton's restitution law in the correct direction, then recombining components. While systematic, it involves multiple coordinate transformations, trigonometry with given tan values, and careful bookkeeping across 13 marks—more demanding than standard M1/M2 collision problems but follows established M4 procedures. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03d Conservation in 2D: vector momentum6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
\includegraphics{figure_1}
Two smooth uniform spheres $A$ and $B$ of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of $A$ is 2.5 m s$^{-1}$ and the speed of $B$ is 1.3 m s$^{-1}$. When they collide the line joining their centres makes an angle $\alpha$ with the direction of motion of $A$ and an angle $\beta$ with the direction of motion of $B$, where $\tan \alpha = \frac{4}{3}$ and $\tan \beta = \frac{12}{5}$ as shown in Fig. 1.
\begin{enumerate}[label=(\alph*)]
\item Find the components of the velocities of $A$ and $B$ perpendicular and parallel to the line of centres immediately before the collision.
[4]
\end{enumerate}
The coefficient of restitution between $A$ and $B$ is $\frac{1}{2}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find, to one decimal place, the speed of each sphere after the collision.
[9]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2004 Q2 [13]}}