4 Two uniform spheres \(A\) and \(B\), of equal radius, are at rest on a smooth horizontal table. Sphere \(A\) has mass \(3 m\) and sphere \(B\) has mass \(m\). Sphere \(A\) is projected directly towards \(B\), with speed \(u\). The coefficient of restitution between the spheres is 0.6 . Find the speeds of \(A\) and \(B\) after they collide. Sphere \(B\) now strikes a wall that is perpendicular to its path, rebounds and collides with \(A\) again. The coefficient of restitution between \(B\) and the wall is \(e\). Given that the second collision between \(A\) and \(B\) brings \(A\) to rest, find \(e\).

## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $3mu_A + mu_B = 3mu$ | M1 | Use conservation of momentum for 1st collision |
| $u_A - u_B = -0.6u$ | M1 A1 | Use Newton's law of restitution (A1 if both correct) |
| $u_A = 0.6u$ and $u_B = 1.2u$ | M1 A1 | Solve for $u_A$ and $u_B$ |
| $u'_B = eu_B\ [= 1.2eu]$ | M1 | Find speed $u'_B$ of $B$ after striking wall |
| $mv_B = 3mu_A - mu'_B$ | B1 | Use conservation of momentum for 2nd collision |
| $v_B = 0.6(u_A + u'_B)$ | B1 | Use Newton's law of restitution |
| $1.6e \times 1.2u = 2.4 \times 0.6u$, $e = \frac{3}{4}$ | M1 A1 | Combine to find $e$ |

**Total: [10]**

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4 Two uniform spheres $A$ and $B$, of equal radius, are at rest on a smooth horizontal table. Sphere $A$ has mass $3 m$ and sphere $B$ has mass $m$. Sphere $A$ is projected directly towards $B$, with speed $u$. The coefficient of restitution between the spheres is 0.6 . Find the speeds of $A$ and $B$ after they collide.

Sphere $B$ now strikes a wall that is perpendicular to its path, rebounds and collides with $A$ again. The coefficient of restitution between $B$ and the wall is $e$. Given that the second collision between $A$ and $B$ brings $A$ to rest, find $e$.

\hfill \mbox{\textit{CAIE FP2 2011 Q4 [10]}}