6.03k Newton's experimental law: direct impact

\includegraphics{figure_5} Two uniform smooth identical spheres \(A\) and \(B\) are moving towards each other on a horizontal surface when they collide. Immediately before the collision \(A\) and \(B\) are moving with speeds \(u_A\) m s\(^{-1}\) and \(u_B\) m s\(^{-1}\) respectively, at acute angles \(\alpha\) and \(\beta\), respectively, to the line of centres. Immediately after the collision \(A\) and \(B\) are moving with speeds \(v_A\) m s\(^{-1}\) and \(v_B\) m s\(^{-1}\) respectively, at right angles and at acute angle \(\gamma\), respectively, to the line of centres (see diagram).

1. Given that \(\sin \beta = 0.96\) and \(\frac{v_B}{u_B} = 1.2\), find the value of \(\sin \gamma\). [2]
2. Given also that, before the collision, the component of \(A\)'s velocity parallel to the line of centres is \(2\) m s\(^{-1}\), find the values of \(u_B\) and \(v_B\). [5]
3. Find the coefficient of restitution between the spheres. [3]
4. Given that the kinetic energy of \(A\) immediately before the collision is \(6.5m\) J, where \(m\) kg is the mass of \(A\), find the value of \(v_A\). [2]

\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]

A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a fixed smooth vertical wall. Immediately before the collision, the speed of \(S\) is \(U\) and its direction of motion makes an angle \(\alpha\) with the wall. The coefficient of restitution between \(S\) and the wall is \(e\). Find the kinetic energy of \(S\) immediately after the collision. [6]

\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]

A small ball is moving on a horizontal plane when it strikes a smooth vertical wall. The coefficient of restitution between the ball and the wall is \(e\). Immediately before the impact the direction of motion of the ball makes an angle of \(60°\) with the wall. Immediately after the impact the direction of motion of the ball makes an angle of \(30°\) with the wall.

1. Find the fraction of the kinetic energy of the ball which is lost in the impact. [6]
2. Find the value of \(e\). [4]

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]

\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]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane] A small smooth ball of mass \(m\) kg is moving on a smooth horizontal plane and strikes a fixed smooth vertical wall. The plane and the wall intersect in a straight line which is parallel to the vector \(2\mathbf{i} + \mathbf{j}\). The velocity of the ball immediately before the impact is \(b\mathbf{i} + \mathbf{j}\) m s\(^{-1}\), where \(b\) is positive. The velocity of the ball immediately after the impact is \(a(\mathbf{i} + \mathbf{j})\) m s\(^{-1}\), where \(a\) is positive.

1. Show that the impulse received by the ball when it strikes the wall is parallel to \((-\mathbf{i} + 2\mathbf{j})\). [1]

Find

1. the coefficient of restitution between the ball and the wall, [8]
2. the fraction of the kinetic energy of the ball that is lost due to the impact. [3]

A small smooth ball of mass \(m\) is falling vertically when it strikes a fixed smooth plane which is inclined to the horizontal at an angle \(\alpha\), where \(0° < \alpha < 45°\). Immediately before striking the plane the ball has speed \(u\). Immediately after striking the plane the ball moves in a direction which makes an angle of \(45°\) with the plane. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(m\), \(u\) and \(e\), the magnitude of the impulse of the plane on the ball. (11)

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]

\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]

A uniform rod \(AB\) of mass \(4m\) is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), through \(A\). The rod is hanging vertically at rest when it is struck at its end \(B\) by a particle of mass \(m\). The particle is moving with speed \(u\), in a direction which is horizontal and perpendicular to \(L\), and after striking the rod it rebounds in the opposite direction with speed \(v\). The coefficient of restitution between the particle and the rod is 1. Show that \(u = 7v\). [7]

Two smooth spheres, \(P\) and \(Q\), of equal radius are free to move on a smooth horizontal surface. The masses of \(P\) and \(Q\) are \(3m\) and \(m\) respectively. \(P\) is set in motion with speed \(u\) directly towards \(Q\), which is initially at rest. \(P\) subsequently collides with \(Q\). \includegraphics{figure_7} Immediately after the collision, \(P\) moves with speed \(v\) and \(Q\) moves with speed \(w\). The coefficient of restitution between the spheres is \(e\).

1. 1. Show that $$v = \frac{u(3-e)}{4}$$ [4 marks]
   2. Find \(w\), in terms of \(e\) and \(u\), simplifying your answer. [2 marks]
2. Deduce that $$\frac{u}{2} \leq v \leq \frac{3u}{4}$$ [2 marks]
3. 1. Find, in terms of \(m\) and \(u\), the maximum magnitude of the impulse that \(P\) exerts on \(Q\). [3 marks]
   2. Describe the impulse that \(Q\) exerts on \(P\). [1 mark]

Two spheres A and B are free to move on a smooth horizontal surface. The masses of A and B are 2 kg and 3 kg respectively. Both A and B are initially at rest. Sphere A is set in motion directly towards sphere B with speed 4 m s\(^{-1}\) and subsequently collides with sphere B The coefficient of restitution between the spheres is \(e\)

1. 1. Show that the speed of B immediately after the collision is $$\frac{8(1 + e)}{5}$$ [4 marks]
   2. Find an expression, in terms of \(e\), for the velocity of A immediately after the collision. [2 marks]
2. It is given that the spheres both move in the same direction after the collision. Find the range of possible values of \(e\) [2 marks]
3. 1. The impulse of sphere A on sphere B is \(I\) The impulse of sphere B on sphere A is \(J\) Given that the collision is perfectly inelastic, find the value of \(I + J\) [1 mark]
   2. State, giving a reason for your answer, whether the value found in part (c)(i) would change if the collision was not perfectly inelastic. [2 marks]

Two spheres, \(A\) and \(B\), of equal size are moving in the same direction along a straight line on a smooth horizontal surface. Sphere \(A\) has mass \(m\) and is moving with speed \(4u\) Sphere \(B\) has mass \(6m\) and is moving with speed \(u\) The diagram shows the spheres and their velocities. \includegraphics{figure_8} Subsequently \(A\) collides directly with \(B\) The coefficient of restitution between \(A\) and \(B\) is \(e\)

1. Find, in terms of \(m\) and \(u\), the total momentum of the spheres before the collision. [1 mark]
2. Show that the speed of \(B\) immediately after the collision is \(\frac{u(3e + 10)}{7}\) [4 marks]
3. After the collision sphere \(A\) moves in the opposite direction. Find the range of possible values for \(e\) [5 marks]

A ball of mass \(m\) kg is held at rest at a height \(h\) metres above a horizontal surface. The ball is released and bounces on the surface. The coefficient of restitution between the ball and the surface is \(e\) Prove that the kinetic energy lost during the first bounce is given by $$mgh(1 - e^2)$$ [4 marks]