Year: 2022
Paper: 3
Question Number: 9
Course: UFM Mechanics
Section: Momentum and Collisions 2
No solution available for this problem.
One question was attempted by well over 90% of the candidates two others by about 90%, and a fourth by over 80%. Two questions were attempted by about half the candidates and a further three questions by about a third of the candidates. Even the other three received attempts from a sixth of the candidates or more, meaning that even the least popular questions were markedly more popular than their counterparts in previous years. Nearly 90% of candidates attempted no more than 7 questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Two particles $A$ and $B$, of masses $m$ and $km$ respectively, lie at rest on a smooth horizontal surface. The coefficient of restitution between the particles is $e$, where $0 < e < 1$. Particle $A$ is then projected directly towards particle $B$ with speed $u$.
Let $v_1$ and $v_2$ be the velocities of particles $A$ and $B$, respectively, after the collision, in the direction of the initial velocity of $A$.
Show that $v_1 = \alpha u$ and $v_2 = \beta u$, where $\alpha = \dfrac{1 - ke}{k+1}$ and $\beta = \dfrac{1+e}{k+1}$.
Particle $B$ strikes a vertical wall which is perpendicular to its direction of motion and a distance $D$ from the point of collision with $A$, and rebounds. The coefficient of restitution between particle $B$ and the wall is also $e$.
Show that, if $A$ and $B$ collide for a second time at a point $\frac{1}{2}D$ from the wall, then
\[ k = \frac{1+e-e^2}{e(2e+1)}\,. \]
\item Three particles $A$, $B$ and $C$, of masses $m$, $km$ and $k^2m$ respectively, lie at rest on a smooth horizontal surface in a straight line, with $B$ between $A$ and $C$. A vertical wall is perpendicular to this line and lies on the side of $C$ away from $A$ and $B$. The distance between $B$ and $C$ is equal to $d$ and the distance between $C$ and the wall is equal to $3d$. The coefficient of restitution between each pair of particles, and between particle $C$ and the wall, is $e$, where $0 < e < 1$. Particle $A$ is then projected directly towards particle $B$ with speed $u$.
Show that, if all three particles collide simultaneously at a point $\frac{3}{2}d$ from the wall, then $e = \frac{1}{2}$.
\end{questionparts}
The most popular of the Applied questions, with a third of candidates attempting it; it was the second-best scoring question on the paper with a mean score of just above half marks. The question relied mainly on the use of conservation of momentum and Newton's experimental law of impact. Most candidates made a very good start with several scoring full or close to full marks in the first part. The difficulties arose later when dealing with the three-particle situation in part (ii). Very few candidates were able to take a step back and see how this problem linked to part (i), resulting in long pages of algebraic manipulation which were inefficient and rarely correct. A good diagram would have made the link so much more obvious!