1 Two smooth spheres \(A\) and \(B\), of equal radii and masses \(2 m\) and \(m\) respectively, lie at rest on a smooth horizontal table. The spheres \(A\) and \(B\) are projected directly towards each other with speeds \(4 u\) and \(3 u\) respectively. The coefficient of restitution between the spheres is \(e\). Find the set of values of \(e\) for which the direction of motion of \(A\) is reversed in the collision.

## Question 1:

| Working/Answer | Mark | Guidance |
|---|---|---|
| $2mv_A + mv_B = 8mu - 3mu$ | B1 | Conservation of momentum |
| $v_A - v_B = -e(4u + 3u)$ | B1 | Restitution (consistent with prev. eqn.) |
| $v_A = \frac{1}{3}(5-7e)u$ | M1 | Solve for $v_A$ (or $3v_A$) |
| $\left[v_B = \frac{1}{3}(5+14e)u\right]$ | | |
| $5 - 7e < 0,\ e > \frac{5}{7}$ or $0.714$ | M1 A1 | Find lower limit on $e$ for which $v_A < 0$ |

**Total: [5]**

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1 Two smooth spheres $A$ and $B$, of equal radii and masses $2 m$ and $m$ respectively, lie at rest on a smooth horizontal table. The spheres $A$ and $B$ are projected directly towards each other with speeds $4 u$ and $3 u$ respectively. The coefficient of restitution between the spheres is $e$. Find the set of values of $e$ for which the direction of motion of $A$ is reversed in the collision.

\hfill \mbox{\textit{CAIE FP2 2014 Q1 [5]}}