1 Two uniform small smooth spheres, \(A\) and \(B\), of equal radii and masses 2 kg and 3 kg respectively, are at rest and not in contact on a smooth horizontal plane. Sphere \(A\) receives an impulse of magnitude 8 N s in the direction \(A B\). The coefficient of restitution between the spheres is \(e\). Find, in terms of \(e\), the speeds of \(A\) and \(B\) after \(A\) collides with \(B\). Given that the spheres move in opposite directions after the collision, show that \(e > \frac { 2 } { 3 }\).

## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $2u_A = 8$, $u_A = 4$ | B1 | |
| $2v_A + 3v_B = 2u_A$ [= 8] | M1 | Conservation of momentum |
| $v_B - v_A = eu_A$ [= 4e] | M1 | Newton's law of restitution |
| $v_A = (2-3e)u_A/5$, $v_B = 2(1+e)u_A/5$ | A1 | AEF |
| $v_A = 4(2-3e)/5$, $v_B = 8(1+e)/5$ | A1 | |
| $2 - 3e < 0$, $e > 2/3$ | B1 | **AG** Use $v_A < 0$ |

**Part marks: 5, 1 | Total: 6**

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1 Two uniform small smooth spheres, $A$ and $B$, of equal radii and masses 2 kg and 3 kg respectively, are at rest and not in contact on a smooth horizontal plane. Sphere $A$ receives an impulse of magnitude 8 N s in the direction $A B$. The coefficient of restitution between the spheres is $e$. Find, in terms of $e$, the speeds of $A$ and $B$ after $A$ collides with $B$.

Given that the spheres move in opposite directions after the collision, show that $e > \frac { 2 } { 3 }$.

\hfill \mbox{\textit{CAIE FP2 2015 Q1 [6]}}