6.03l Newton's law: oblique impacts

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_5} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2m\) kg and \(m\) kg respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, \(A\) is moving with speed \(a\text{ ms}^{-1}\) in a direction making an angle \(\alpha\) with the line of centres and \(B\) is moving towards \(A\) with speed \(b\text{ ms}^{-1}\) in a direction making an angle \(\beta\) with the line of centres (see diagram). After the collision, \(A\) moves with velocity \(2\text{ ms}^{-1}\) in a direction perpendicular to the line of centres and \(B\) moves with velocity \(2\text{ ms}^{-1}\) in a direction making an angle of \(45°\) with the line of centres. The coefficient of restitution between \(A\) and \(B\) is \(\frac{2}{3}\).

1. Show that \(a\cos \alpha = \frac{5}{3}\sqrt{2}\) and find \(b\cos \beta\). [7]
2. Find the values of \(a\) and \(\alpha\). [4]

\includegraphics{figure_3} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2m\) kg and \(3m\) kg respectively. The spheres are approaching each other on a horizontal surface when they collide. Before the collision \(A\) is moving with speed \(5\) m s\(^{-1}\) in a direction making an angle \(\alpha\) with the line of centres, where \(\cos \alpha = \frac{4}{5}\), and \(B\) is moving with speed \(3\frac{1}{4}\) m s\(^{-1}\) in a direction making an angle \(\beta\) with the line of centres, where \(\cos \beta = \frac{5}{13}\). A straight vertical wall is situated to the right of \(B\), perpendicular to the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is \(\frac{2}{5}\).

1. Find the speed of \(A\) after the collision. Find also the component of the velocity of \(B\) along the line of centres after the collision. [7]

\(B\) subsequently hits the wall.

1. Explain why \(A\) and \(B\) will have a second collision if the coefficient of restitution between \(B\) and the wall is sufficiently large. Find the set of values of the coefficient of restitution for which this second collision will occur. [3]

A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta\), \(\theta < 45°\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.

1. Show that \(e = \tan^2 \theta\). [6]
2. Show that the magnitude of the impulse exerted by \(P\) on the plane is \(mu \sec \theta\). [4]

\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.] The vector \(\mathbf{n} = (-\frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j})\) and the vector \(\mathbf{p} = (-\frac{4}{5}\mathbf{i} + \frac{3}{5}\mathbf{j})\) are perpendicular unit vectors.

1. Verify that \(\frac{3}{5}\mathbf{n} + \frac{4}{5}\mathbf{p} = (\mathbf{i} + 3\mathbf{j})\). [2]

A smooth uniform sphere \(S\) of mass 0.5 kg is moving on a smooth horizontal plane when it collides with a fixed vertical wall which is parallel to \(\mathbf{p}\). Immediately after the collision the velocity of \(S\) is \((\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\). The coefficient of restitution between \(S\) and the wall is \(\frac{3}{5}\).

1. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the velocity of \(S\) immediately before the collision. [5]
2. Find the energy lost in the collision. [3]

A small smooth sphere \(S\) of mass \(m\) is attached to one end of a light inextensible string of length \(2a\). The other end of the string is attached to a fixed point \(A\) which is at a distance \(a\sqrt{3}\) from a smooth vertical wall. The sphere \(S\) hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed \(\sqrt{\left(\frac{37ga}{5}\right)}\).

1. Show that \(S\) strikes the wall with speed \(\sqrt{\left(\frac{27ga}{5}\right)}\). [4] Given that the loss in kinetic energy due to the impact with the wall is \(\frac{3mga}{5}\),
2. find the coefficient of restitution between \(S\) and the wall. [7]

Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(km\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of \(60°\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac{3u}{4(k + 1)}\). [6] Immediately after the collision the direction of motion of \(A\) makes an angle arctan \((2\sqrt{3})\) with the direction of motion of \(B\).
2. Show that \(k = \frac{1}{2}\). [6]
3. Find the loss of kinetic energy due to the collision. [4]

A small smooth ball of mass \(\frac{1}{2}\) kg is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{1}{3}\). Immediately before striking the plane the ball has speed 10 m s\(^{-1}\). The coefficient of restitution between ball and plane is \(\frac{1}{2}\). Find

1. the speed, to 3 significant figures, of the ball immediately after the impact, [5]
2. the magnitude of the impulse received by the ball as it strikes the plane. [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]

A smooth uniform sphere \(S\) is moving on a smooth horizontal plane when it collides obliquely with an identical sphere \(T\) which is at rest on the plane. Immediately before the collision \(S\) is moving with speed \(U\) in a direction which makes an angle of \(60°\) with the line joining the centres of the spheres. The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(e\) and \(U\) where necessary,
   1. the speed and direction of motion of \(S\) immediately after the collision,
   2. the speed and direction of motion of \(T\) immediately after the collision.
   (12)

The angle through which the direction of motion of \(S\) is deflected is \(\delta°\).

1. Find
   1. the value of \(e\) for which \(\delta\) takes the largest possible value,
   2. the value of \(\delta\) in this case.
   (3)

A small ball is moving on a smooth horizontal plane when it collides obliquely with a smooth plane vertical wall. The coefficient of restitution between the ball and the wall is \(\frac{1}{3}\). The speed of the ball immediately after the collision is half the speed of the ball immediately before the collision. Find the angle through which the path of the ball is deflected by the collision. [8]

\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]

A sphere, of mass 0.2 kg, moving on a smooth horizontal surface, collides with a fixed wall. Before the collision the sphere moves with speed 5 m s\(^{-1}\) at an angle of 60° to the wall. After the collision the sphere moves with speed \(v\) m s\(^{-1}\) at an angle of \(\theta\)° to the wall. The velocities are shown in the diagram below. \includegraphics{figure_7} The coefficient of restitution between the wall and the sphere is 0.7

1. Assume that the wall is smooth.
   1. Find the value of \(v\) Give your answer to two significant figures. [4 marks]
   2. Find the value of \(\theta\) Give your answer to the nearest whole number. [2 marks]
   3. Find the magnitude of the impulse exerted on the sphere by the wall. Give your answer to two significant figures. [2 marks]
2. In reality the wall is not smooth. Explain how this would cause a change in the magnitude of the impulse calculated in part (a)(iii). [2 marks]

Two smooth circular discs \(A\) and \(B\) are moving on a smooth horizontal plane when they collide. The mass of \(A\) is \(5\) kg and the mass of \(B\) is \(3\) kg. At the instant before they collide, • the velocity of \(A\) is \(4\) m s\(^{-1}\) at an angle of \(60°\) to the line of centres, • the velocity of \(B\) is \(6\) m s\(^{-1}\) along the line of centres (see diagram). \includegraphics{figure_3} The coefficient of restitution for collisions between the two discs is \(\frac{3}{4}\). Determine the angle that the velocity of \(A\) makes with the line of centres after the collision. [7]

A small ball of mass \(m\) kg is held at a height of \(78.4\) m above horizontal ground. The ball is released from rest, falls vertically and rebounds from the ground. The coefficient of restitution between the ball and ground is \(e\). The ball continues to bounce until it comes to rest after \(6\) seconds.

1. Determine the value of \(e\). [8]
2. Given that the magnitude of the impulse that the ground exerts on the ball at the first bounce is \(23.52\) Ns, determine the value of \(m\). [2]

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]

Two objects, \(A\) of mass 18 kg and \(B\) of mass 7 kg, are moving in the same straight line on a smooth horizontal surface. Initially, they are moving with the same speed of \(4\text{ ms}^{-1}\) and in the same direction. Object \(B\) collides with a vertical wall which is perpendicular to its direction of motion and rebounds with a speed of \(3\text{ ms}^{-1}\). Subsequently, the two objects \(A\) and \(B\) collide directly. The coefficient of restitution between the two objects is \(\frac{5}{7}\).

1. Find the coefficient of restitution between \(B\) and the wall. [1]
2. Determine the speed of \(A\) and the speed of \(B\) immediately after the two objects collide. [7]
3. Calculate the impulse exerted by \(A\) on \(B\) due to the collision and clearly state its units. [2]
4. Find the loss in energy due to the collision between \(A\) and \(B\). [2]
5. State the direction of motion of \(A\) relative to the wall after the collision with \(B\). [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]