2 A small uniform sphere \(A\), of mass \(2 m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\). Find expressions for the speeds of \(A\) and \(B\) immediately after the collision. Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is 0.4 . After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal. Find \(e\). Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\).
\(3 A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(A B = 3 a \mathrm {~m}\). One end of a light elastic string, of natural length \(a \mathrm {~m}\) and modulus of elasticity \(m g \mathrm {~N}\), is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). One end of a second light elastic string, of natural length \(k a m\) and modulus of elasticity \(2 m g \mathrm {~N}\), is attached to \(B\). The other end of this string is attached to \(P\). Given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(A B\), find the value of \(k\). The particle \(P\) is released from rest at a point between \(A\) and \(B\) where both strings are taut. Show that \(P\) performs simple harmonic motion and state the period of the motion. In the case where \(P\) is released from rest at a distance \(0.2 a \mathrm {~m}\) from \(M\), the speed of \(P\) is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(P\) is \(0.05 a \mathrm {~m}\) from \(M\). Find the value of \(a\).