Oblique collision of spheres

\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision \(A\)'s direction of motion makes an angle of \(\alpha°\) with the line of centres, and \(B\)'s direction of motion is perpendicular to that of \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\). Immediately after the collision, \(B\) moves in a direction at right angles to the line of centres.

1. Show that \(\tan \alpha = \frac{1+e}{1-e}\). [4]

\includegraphics{figure_6} Two uniform smooth spheres A and B of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle \(\alpha\) with the line of centres, and B's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{1}{3}\) and \(2\cos\beta = \cos\alpha\).

1. Show that the direction of motion of A after the collision is perpendicular to the line of centres. [4]

The total kinetic energy of the spheres after the collision is \(\frac{3}{4}mu^2\).

1. Find the value of \(\alpha\). [4]

\includegraphics{figure_6} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(km\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides with sphere \(B\) which is at rest. Immediately before the collision, \(A\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{1}{3}\).

1. Show that the speed of \(B\) after the collision is \(\frac{4u \cos \theta}{3(1 + k)}\). [3]

Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(km\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Show that the speed of \(B\) after the collision is \(\frac{3u \cos \alpha}{2(1 + k)}\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k\), \(u\) and \(\alpha\). [4]

After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).

1. Given that \(\tan \alpha = \frac{2}{3}\), find the possible values of \(k\). [5]

\includegraphics{figure_4} Two identical smooth uniform spheres \(A\) and \(B\) each have mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(2u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(30°\) with the line of centres, and \(B\)'s direction of motion is perpendicular to the line of centres (see diagram). After the collision, \(A\) and \(B\) are moving in the same direction. The coefficient of restitution between the spheres is \(e\).

1. Find the value of \(e\). [5]
2. Find the loss in the total kinetic energy of the spheres as a result of the collision. [3]

\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{2}{3}m\) respectively. The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle of \(60°\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{2}{3}\).

1. Find the angle through which the direction of motion of \(B\) is deflected by the collision. [6]
2. Find the loss in the total kinetic energy of the system as a result of the collision. [3]

\includegraphics{figure_7} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{1}{2}m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{5}{8}\) and \(\alpha + \beta = 90°\).

1. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\). [4]

The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.

1. Find the value of \(\tan \alpha\). [5]

\includegraphics{figure_1} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving with equal speeds \(u\) on a smooth horizontal surface when they collide. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(60°\) with the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(e\). After the collision, the component of the velocity of \(A\) along the line of centres is \(v\) and \(B\) moves perpendicular to the line of centres. Sphere \(A\) now has twice as much kinetic energy as sphere \(B\).

1. Show that \(v = \frac{1}{2}u(4\cos\theta - 1)\). [1]
2. Find the value of \(\cos\theta\). [4]
3. Find the value of \(e\). [2]

\includegraphics{figure_4} Two smooth vertical walls meet at right angles. The smooth sphere \(A\), with mass \(m\), is at rest on a smooth horizontal surface and is at a distance \(d\) from each wall. An identical smooth sphere \(B\) is moving on the horizontal surface with speed \(u\) at an angle \(\theta\) with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Find the value of \(\tan \theta\). [4]
2. Find the percentage loss in the total kinetic energy of the spheres as a result of this collision. [3]

In this question use \(\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}\). A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60°\) with the wall, as shown in the diagram. \includegraphics{figure_6} The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac{1}{4}u(1 + e)\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision. [7 marks]
2. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac{1 + e}{7 - e}\). [7 marks]

\includegraphics{figure_3} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(4\) kg and \(2\) kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \text{ m s}^{-1}\). The spheres are moving in opposite directions, each at \(60°\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.

1. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres. [8]
2. Find the coefficient of restitution between the spheres. [2]

\includegraphics{figure_1} A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\). Show that \((1-e) \tan \beta = 2 \tan \alpha\). [11]

\includegraphics{figure_2} Two small smooth spheres \(A\) and \(B\), of equal size and of mass \(m\) and \(2m\) respectively, are moving initially with the same speed \(U\) on a smooth horizontal floor. The spheres collide when their centres are on a line \(L\). Before the collision the spheres are moving towards each other, with their directions of motion perpendicular to each other and each inclined at an angle of \(45°\) to the line \(L\), as shown in Figure 2. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Find the magnitude of the impulse which acts on \(A\) in the collision. [9]

\includegraphics{figure_3} The line \(L\) is parallel to and a distance \(d\) from a smooth vertical wall, as shown in Figure 3.

1. Find, in terms of \(d\), the distance between the points at which the spheres first strike the wall. [5]

\includegraphics{figure_2} Two smooth uniform spheres \(A\) and \(B\), of equal radius \(r\), have masses \(3m\) and \(2m\) respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision they are moving with speeds \(u\) and \(2u\) respectively. The centres of the spheres are moving towards each other along parallel paths at a distance \(1.6r\) apart, as shown in Figure 2. The coefficient of restitution between the two spheres is \(\frac{1}{6}\). Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision. [10]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(m\) and the mass of \(B\) is \(3m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, \(A\) is moving with speed \(3u\) at angle \(\alpha\) to the line of centres and \(B\) is moving with speed \(u\) at angle \(\beta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the two spheres is \(\frac{1}{5}\). It is given that \(\cos \alpha = \frac{1}{3}\) and \(\cos \beta = \frac{2}{3}\) and that \(\alpha\) and \(\beta\) are both acute angles.

1. Find the magnitude of the impulse on \(A\) due to the collision in terms of \(m\) and \(u\). [8]
2. Express the kinetic energy lost by \(A\) in the collision as a fraction of its initial kinetic energy. [4]

\includegraphics{figure_2} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 3 kg and velocity (2\(\mathbf{i}\) + \(\mathbf{j}\)) m s\(^{-1}\), and sphere \(B\) has mass 5 kg and velocity (\(-\mathbf{i}\) + \(\mathbf{j}\)) m s\(^{-1}\). When the spheres collide the line joining their centres is parallel to \(\mathbf{i}\), as shown in Fig. 2. Given that the direction of \(A\) is deflected through a right angle by the collision, find

1. the velocity of \(A\) after the collision, [5]
2. the coefficient of restitution between the spheres. [6]

Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u_1 \mathbf{i} + u_2 \mathbf{j}\) towards B, which is at rest. The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf{i}\). The coefficient of restitution between A and B is \(e\).

1. 1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf{i}\). [1]
   2. Determine this impulse in terms of \(\lambda\), \(m\), \(e\) and \(u_1\). [6]

The loss in kinetic energy due to the collision between A and B is \(\frac{1}{8}mu_1^2\).

1. Determine the range of possible values of \(\lambda\). [6]

Two small uniform discs A and B, of equal radius, have masses 3 kg and 5 kg respectively. The discs are sliding on a smooth horizontal surface and collide obliquely. The contact between the discs is smooth and A is stationary after the collision. Immediately before the collision B is moving with speed \(2 \text{ m s}^{-1}\) in a direction making an angle of \(60°\) with the line of centres, XY (see diagram below). \includegraphics{figure_12}

1. Explain how you can tell that A must have been moving along XY before the collision. [1]

The coefficient of restitution between A and B is 0.8.

1. • Determine the speed of A immediately before the collision. • Determine the speed of B immediately after the collision. [7]
2. Determine the angle turned through by the direction of B in the collision. [3]

Disc B subsequently collides with a smooth wall, which is parallel to XY. The kinetic energy of B after the collision with the wall is 95% of the kinetic energy of B before the collision with the wall.

1. Determine the coefficient of restitution between B and the wall. [4]

Two uniform small smooth spheres A and B have equal radii and equal masses. The spheres are on a smooth horizontal surface. Sphere A is moving at an acute angle \(\alpha\) to the line of centres, when it collides with B, which is stationary. After the impact A is moving at an acute angle \(\beta\) to the line of centres. The coefficient of restitution between A and B is \(\frac{1}{3}\).

1. Show that \(\tan\beta = 3\tan\alpha\). [5]
2. Explain why the assumption that the contact between the spheres is smooth is needed in answering part (a). [1] It is given that A is deflected through an angle \(\gamma\).
3. Determine, in terms of \(\alpha\), an expression for \(\tan\gamma\). [2]
4. Determine the maximum value of \(\gamma\). You do not need to justify that this value is a maximum. [5]

\includegraphics{figure_9} Fig. 9 shows the instant of impact of two identical uniform smooth spheres, A and B, each with mass \(m\). Immediately before they collide, the spheres are sliding towards each other on a smooth horizontal table in the directions shown in the diagram, each with speed \(v\). The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Show that, immediately after the collision, the speed of A is \(\frac{1}{8}v\). Find its direction of motion. [6]
2. Find the percentage of the original kinetic energy that is lost in the collision. [7]
3. State where in your answer to part (i) you have used the assumption that the contact between the spheres is smooth. [1]

\includegraphics{figure_4} Two uniform smooth spheres A and B of equal radius are moving on a horizontal surface when they collide. A has mass 0.1 kg and B has mass 0.4 kg. Immediately before the collision A is moving with speed 2.8 ms\(^{-1}\) along the line of centres, and B is moving with speed 1 ms\(^{-1}\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision A is stationary. Find:

1. the coefficient of restitution between A and B, [5]
2. the angle turned through by the direction of motion of B as a result of the collision. [4]

\includegraphics{figure_7} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{1}{2}m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{2}{3}\) and \(\alpha + \beta = 90°\).

1. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\). [4]

The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.

1. Find the value of \(\tan \alpha\). [5]

\includegraphics{figure_8} A smooth sphere with centre \(A\) and of mass 2 kg, moving at 13 m s\(^{-1}\) on a smooth horizontal plane, strikes a smooth sphere with centre \(B\) and of mass 3 kg moving at 5 m s\(^{-1}\) on the same smooth horizontal plane. The spheres have equal radii. The directions of motion immediately before impact are at angles \(\tan^{-1}\left(\frac{2}{13}\right)\) to \(\overrightarrow{AB}\) and \(\tan^{-1}\left(\frac{4}{3}\right)\) to \(\overrightarrow{BA}\) respectively (see diagram). Given that the coefficient of restitution is \(\frac{2}{3}\), find the speeds of the spheres after impact. [9]

A smooth sphere \(A\) of mass \(m\) is projected with speed \(u\) along a smooth horizontal surface and strikes a stationary smooth sphere \(B\) of equal radius but of mass \(M\). The direction of motion of \(A\) before the impact makes an acute angle \(\theta\) with the line of centres at the moment of contact. After the impact, the direction of motion of \(A\) is perpendicular to the initial direction of motion of \(A\). The coefficient of restitution between the two spheres is \(e\). Given that \(Me \geq m\), prove that $$\tan^2 \theta = \frac{Me - m}{m + M}.$$ [8]