Two-sphere oblique collision

2 A smooth uniform ball travelling along a smooth horizontal table collides with a second smooth uniform ball of the same mass and radius which is at rest on the table. At the moment of impact the line of centres makes an angle of \(30 ^ { \circ }\) with the direction in which the first ball is moving. If the coefficient of restitution between the balls is \(e\), show that

1. the component of the first ball's velocity, along the line of centres, after the impact is $$\frac { \sqrt { 3 } u } { 4 } ( 1 - e )$$
2. the first ball is deflected by the impact through an angle \(\theta\), where $$\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }$$

A uniform smooth sphere \(A\) moves on a smooth horizontal surface towards a smooth vertical wall. Immediately before the sphere hits the wall it has components of velocity parallel and perpendicular to the wall each of magnitude \(4\) m s\(^{-1}\). Immediately after hitting the wall the components have magnitudes \(u\) m s\(^{-1}\) and \(v\) m s\(^{-1}\), respectively (see Fig. 1). \includegraphics{figure_1}

1. Given that the coefficient of restitution between the sphere and the wall is \(\frac{1}{4}\), state the values of \(u\) and \(v\). [2]

Shortly after hitting the wall the sphere \(A\) comes into contact with another uniform smooth sphere \(B\), which has the same mass and radius as \(A\). The sphere \(B\) is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic. \includegraphics{figure_2}

1. Find, for each sphere, its speed and its direction of motion immediately after the contact. [6]

\includegraphics{figure_1} A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal table. The sphere \(S\) collides with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(km\), \(k > 1\), which is at rest on the table. The coefficient of restitution between the spheres is \(e\). Immediately before the spheres collide the direction of motion of \(S\) makes an angle \(\theta\) with the line joining their centres, as shown in Fig. 1. Immediately after the collision the directions of motion of \(S\) and \(T\) are perpendicular.

1. Show that \(e = \frac{1}{k}\). [6]

Given that \(k = 2\) and that the kinetic energy lost in the collision is one quarter of the initial kinetic energy,

1. find the value of \(\theta\). [6]

\includegraphics{figure_2} A small ball \(Q\) of mass \(2m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac{13}{12}u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d_1\) from \(B\), and \(BC\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d_2\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(BC\), as shown in Fig. 2, where \(\tan \alpha = \frac{5}{12}\). The line of centres of \(P\) and \(Q\) is parallel to \(BC\). After the collision \(Q\) moves towards \(C\) with speed \(\frac{5}{4}u\).

1. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(CB\) are \(\frac{1}{4}u\) and \(\frac{5}{12}u\) respectively. [4]
2. Find the coefficient of restitution between \(P\) and \(Q\). [2]
3. Show that when \(Q\) reaches \(C\), \(P\) is at a distance \(\frac{4}{5}d_1\) from \(W\). [3]

For each collision between a ball and a wall the coefficient of restitution is \(\frac{1}{2}\). Given that the balls collide with each other again,

1. show that the time between the two collisions of the balls is \(\frac{15d_1}{u}\). [4]
2. find the ratio \(d_1 : d_2\). [5]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of \(A\) is 2.5 m s\(^{-1}\) and the speed of \(B\) is 1.3 m s\(^{-1}\). When they collide the line joining their centres makes an angle \(\alpha\) with the direction of motion of \(A\) and an angle \(\beta\) with the direction of motion of \(B\), where \(\tan \alpha = \frac{4}{3}\) and \(\tan \beta = \frac{12}{5}\) as shown in Fig. 1.

1. Find the components of the velocities of \(A\) and \(B\) perpendicular and parallel to the line of centres immediately before the collision. [4]

The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).

1. Find, to one decimal place, the speed of each sphere after the collision. [9]

\includegraphics{figure_3} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point \(X\) with a stationary van of mass 800 kg. After the collision the van came to rest at the point \(A\) having travelled a horizontal distance of 45 m, and the car came to rest at the point \(B\) having travelled a horizontal distance of 21 m. The angle \(AXB\) is 90°. The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.

1. Find the coefficient of restitution between the car and the van. [5]

The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along \(AX\) and the car to a constant horizontal force of 300 N along \(BX\).

1. Find the speed of the car immediately before the collision. [9]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors.] Two smooth uniform spheres \(A\) and \(B\) have equal radius but masses \(m\) and \(5m\) respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocities of \(A\) and \(B\) are \((\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and \((-\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\) respectively. Immediately after the collision, the velocity of \(A\) is \((-2\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\).

1. By considering the impulse on \(A\), find a unit vector parallel to the line joining the centres of the spheres when they collide. [4]
2. Find the velocity of \(B\) immediately after the collision. [3]

\includegraphics{figure_11} The diagram shows two small identical uniform smooth spheres, A and B, just before A collides with B. Sphere B is at rest on a horizontal table with its centre vertically above the edge of the table. Sphere A is projected vertically upwards so that, just before it collides with B, the speed of A is \(U\) m s\(^{-1}\) and it is in contact with the vertical side of the table. The point of contact of A with the vertical side of the table and the centres of the spheres are in the same vertical plane.

1. Show that on impact the line of centres makes an angle of 30° with the vertical. [1]

The coefficient of restitution between A and B is \(\frac{1}{2}\). After the impact B moves freely under gravity.

1. Determine, in terms of \(U\) and \(g\), the time taken for B to first return to the table. [7]