Year: 2024
Paper: 3
Question Number: 9
Course: LFM Pure and Mechanics
Section: Momentum and Collisions
No solution available for this problem.
The total entry was an increase on that of 2023 by more than 10%. One question was attempted by more than 98% of candidates, another two by about 80%, and another five by between 50% and 70%. The remaining four questions were attempted by between 5% and 30% of candidates, these being from Section B: Mechanics, and Section C: Probability and Statistics, though the Statistics questions were in general attempted more often and more successfully. All questions were perfectly solved by some candidates. About 84% of candidates attempted no more than 7 questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
The origin $O$ of coordinates lies on a smooth horizontal table and the $x$- and $y$-axes lie in the plane of the table. A smooth sphere $A$ of mass $m$ and radius $r$ is at rest on the table with its lowest point at the origin.
A second smooth sphere $B$ has the same mass and radius and also lies on the table. Its lowest point has $y$-coordinate $2r\sin\alpha$, where $\alpha$ is an acute angle, and large positive $x$-coordinate.
Sphere $B$ is now projected parallel to the $x$-axis, with speed $u$, so that it strikes sphere $A$. The coefficient of restitution in this collision is $\frac{1}{3}$.
\begin{questionparts}
\item Show that, after the collision, sphere $B$ moves with velocity
\[\begin{pmatrix} -\frac{1}{3}u\bigl(1 + 2\sin^2\alpha\bigr) \\ \frac{2}{3}u\sin\alpha\cos\alpha \end{pmatrix}.\]
\item Show further that the lowest point of sphere $B$ crosses the $y$-axis at the point $(0, Y)$, where $Y = 2r(\cos\alpha\tan\beta + \sin\alpha)$ and
\[\tan\beta = \frac{2\sin\alpha\cos\alpha}{1 + 2\sin^2\alpha}.\]
\end{questionparts}
A third sphere $C$ of radius $r$ is at rest with its lowest point at $(0, h)$ on the table, where $h > 0$.
\begin{questionparts}
\setcounter{enumi}{2}
\item Show that, if $h > Y + 2r\sec\beta$, sphere $B$ will not strike sphere $C$ in its motion after the collision with sphere $A$.
\item Show that $Y < 2r\sec\beta$.
Hence show that sphere $B$ will not strike sphere $C$ for any value of $\alpha$, if $h > \dfrac{8r}{\sqrt{3}}$.
\end{questionparts}
This was an unpopular question, only being attempted by about a seventh of the candidates. It was also the second least successful with a mean score of only 4/20. There were mixed responses, and it mostly depended on how the diagram was set up, that is in which directions candidates chose to label the velocities. Many candidates struggled to understand how to apply the restitution law when the particles collide obliquely rather than directly along the line of centres. Some tried to use total speeds of the particles rather than the speeds along the line of contact, and some tried to use the horizontal speeds. Many also did not use vectors correctly, drawing vectors in certain directions then not introducing necessary negative signs. Other than that, part (i) was done well and most understood how to rotate the solution back into usual x-y directions. Those who got to part (ii) generally did it easily. Most found part (iii) trickier, and it tended to be either done well or not really started. Once the diagram was set up, it was found to be straightforward, and most who got that far saw how to proceed. There were very few significant attempts at part (iv).