Impulse and momentum (advanced)

\includegraphics{figure_3} Two perfectly elastic small smooth spheres \(A\) and \(B\) have masses \(3m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane with \(B\) at a distance \(a\) from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and \(B\) is between \(A\) and the barrier (see diagram). Sphere \(A\) is projected towards sphere \(B\) with speed \(u\) and, after the collision between the spheres, \(B\) hits the barrier. The coefficient of restitution between \(B\) and the barrier is \(\frac{1}{4}\). Find the speeds of \(A\) and \(B\) immediately after they first collide, and the distance from the barrier of the point where they collide for the second time. [9]

\includegraphics{figure_2} A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. It collides at an angle \(\alpha\) with a fixed smooth vertical barrier. After the collision, \(P\) moves at an angle \(\theta\) with the barrier, where \(\tan\theta = \frac{1}{3}\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(e\). The particle \(P\) loses 20% of its kinetic energy as a result of the collision. Find the value of \(e\). [5]

Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). \includegraphics{figure_1} Find the value of \(\tan\theta\). [6]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). Find the value of \(\tan\theta\). [6]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(\frac{1}{2}u\) respectively. Immediately before the collision, \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). As a result of the collision, the direction of motion of \(A\) is reversed and its speed is reduced to \(\frac{1}{4}u\). The direction of motion of \(B\) again makes an angle \(\theta\) with the line of centres, but on the opposite side of the line of centres. The speed of \(B\) is unchanged. Find the value of the coefficient of restitution between the spheres. [4]

\includegraphics{figure_3} The diagram shows two identical smooth uniform spheres \(A\) and \(B\) of equal radii and each of mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(2u\) and \(3u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle \(\theta\) with the line of centres and \(B\)'s direction of motion is perpendicular to that of \(A\). After the collision, \(B\) moves perpendicular to the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{3}\).

1. Find the value of \(\tan \theta\). [3]
2. Find the total loss of kinetic energy as a result of the collision. [2]
3. Find, in degrees, the angle through which the direction of motion of \(A\) is deflected as a result of the collision. [2]

\includegraphics{figure_6} A trapeze artiste of mass 60 kg is attached to the end \(A\) of a light inextensible rope \(OA\) of length 5 m. The artiste must swing in an arc of a vertical circle, centre \(O\), from a platform \(P\) to another platform \(Q\), where \(PQ\) is horizontal. The other end of the rope is attached to the fixed point \(O\) which lies in the vertical plane containing \(PQ\), with \(\angle POQ = 120^{\circ}\) and \(OP = OQ = 5\) m, as shown in Figure 6. As part of her act, the artiste projects herself from \(P\) with speed \(\sqrt{15}\) m s\(^{-1}\) in a direction perpendicular to the rope \(OA\) and in the plane \(POQ\). She moves in a circular arc towards \(Q\). At the lowest point of her path she catches a ball of mass \(m\) kg which is travelling towards her with speed 3 m s\(^{-1}\) and parallel to \(QP\). After catching the ball, she comes to rest at the point \(Q\). By modelling the artiste and the ball as particles and ignoring her air resistance, find

1. the speed of the artiste immediately before she catches the ball, [4]
2. the value of \(m\), [7]
3. the tension in the rope immediately after she catches the ball. [3]

A smooth uniform sphere \(A\) has mass \(2m\) kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(m\) kg. The spheres are moving on a smooth horizontal plane when they collide. At the instant of collision the line joining the centres of the spheres is parallel to \(\mathbf{j}\). Immediately after the collision, the velocity of \(A\) is \((3\mathbf{i} - \mathbf{j})\) m s\(^{-1}\) and the velocity of \(B\) is \((2\mathbf{i} + \mathbf{j})\) m s\(^{-1}\). The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Find the velocities of the two spheres immediately before the collision. [7]
2. Find the magnitude of the impulse in the collision. [2]
3. Find, to the nearest degree, the angle through which the direction of motion of \(A\) is deflected by the collision. [4]

The diagram below shows a particle \(P\), of mass 2.5 kg, attached by means of two light inextensible strings fixed at points \(A\) and \(B\). Point \(A\) is vertically above point \(B\). \(BP\) makes an angle of \(60°\) with the upward vertical and \(AP\) is inclined at an angle \(\theta\) to the downward vertical where \(\cos\theta = 0.8\). The particle \(P\) describes a horizontal circle with constant angular speed \(\omega\) radians per second about centre \(C\) with both strings taut. \includegraphics{figure_7} The tension in the string \(BP\) is 39.2 N.

1. Calculate the tension in the string \(AP\). [4]
2. Given that the length of the string \(AP\) is 1.5 m, find the value of \(\omega\). [5]
3. Calculate the kinetic energy of \(P\). [3]

The diagram below shows two spheres \(A\) and \(B\), of equal radii, moving in the same direction on a smooth horizontal surface. Sphere \(A\), of mass \(3\) kg, is moving with speed \(4\) ms\(^{-1}\) and sphere \(B\), of mass \(2\) kg, is moving with speed \(10\) ms\(^{-1}\). \includegraphics{figure_5} Sphere \(B\) is then given an impulse after which it moves in the opposite direction with speed \(6\) ms\(^{-1}\).

1. Calculate the magnitude and direction of the impulse exerted on \(B\). [3]

Sphere \(B\) continues to move with speed \(6\) ms\(^{-1}\) so that it collides directly with sphere \(A\). The kinetic energy lost due to the collision is \(45\) J.

1. Calculate the speed of \(A\) and the speed of \(B\) immediately after the two spheres collide. State the direction in which each sphere is moving relative to its motion immediately before the collision. [8]