6.03k Newton's experimental law: direct impact

6. \begin{figure}[h] \end{figure} A small smooth uniform sphere \(S\) is at rest on a smooth horizontal floor at a distance \(d\) from a straight vertical wall. An identical sphere \(T\) is projected along the floor with speed \(U\) towards \(S\) and in a direction which is perpendicular to the wall. At the instant when \(T\) strikes \(S\) the line joining their centres makes an angle \(\alpha\) with the wall, as shown in Fig. 3. Each sphere is modelled as having negligible diameter in comparison with \(d\). The coefficient of restitution between the spheres is \(e\).

1. Show that the components of the velocity of \(T\) after the impact, parallel and perpendicular to the line of centres, are \(\frac { 1 } { 2 } U ( 1 - e ) \sin \alpha\) and \(U \cos \alpha\) respectively.
2. Show that the components of the velocity of \(T\) after the impact, parallel and perpendicular to the wall, are \(\frac { 1 } { 2 } U ( 1 + e ) \cos \alpha \sin \alpha\) and \(\frac { 1 } { 2 } U \left[ 2 - ( 1 + e ) \sin ^ { 2 } \alpha \right]\) respectively. The spheres \(S\) and \(T\) strike the wall at the points \(A\) and \(B\) respectively.
   Given that \(e = \frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\),
3. find, in terms of \(d\), the distance \(A B\). \section*{END}

2. \begin{figure}[h] \end{figure} A smooth uniform sphere \(P\) is at rest on a smooth horizontal plane, when it is struck by an identical sphere \(Q\) moving on the plane. Immediately before the impact, the line of motion of the centre of \(Q\) is tangential to the sphere \(P\), as shown in Fig. 1. The direction of motion of \(Q\) is turned through \(30 ^ { \circ }\) by the impact. Find the coefficient of restitution between the spheres.

4. \begin{figure}[h] \end{figure} A small smooth ball \(B\), moving on a horizontal plane, collides with a fixed vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(2 \theta\), where \(0 ^ { \circ } < \theta < 45 ^ { \circ }\). Immediately after the collision the angle between the direction of motion of \(B\) and the wall is \(\theta\), as shown in Figure 1. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 8 }\), find the value of \(\tan \theta\).
(8)

1. \begin{figure}[h] \end{figure} A fixed smooth plane is inclined to the horizontal at an angle of \(45 ^ { \circ }\). A particle \(P\) is moving horizontally and strikes the plane. Immediately before the impact, \(P\) is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, \(P\) is moving in a direction which makes an angle of \(30 ^ { \circ }\) with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of \(P\) which is lost in the impact.

5. Two small smooth spheres \(A\) and \(B\), of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that the velocity of \(B\) immediately after the collision is \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the impulse of \(B\) on \(A\) in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to \(\mathbf { i } + \mathbf { j }\).
3. Find the coefficient of restitution between \(A\) and \(B\).
   \section*{June 2009}

2. Two smooth uniform spheres \(S\) and \(T\) have equal radii. The mass of \(S\) is 0.3 kg and the mass of \(T\) is 0.6 kg . The spheres are moving on a smooth horizontal plane and collide obliquely. Immediately before the collision the velocity of \(S\) is \(\mathbf { u } _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(T\) is \(\mathbf { u } _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the spheres is 0.5 . Immediately after the collision the velocity of \(S\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(T\) is \(( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Given that when the spheres collide the line joining their centres is parallel to \(\mathbf { i }\),

1. find
   1. \(\mathbf { u } _ { 1 }\),
   2. \(\mathbf { u } _ { 2 }\). After the collision, \(T\) goes on to collide with a smooth vertical wall which is parallel to \(\mathbf { j }\). Given that the coefficient of restitution between \(T\) and the wall is also 0.5 , find
2. the angle through which the direction of motion of \(T\) is deflected as a result of the collision with the wall,
3. the loss in kinetic energy of \(T\) caused by the collision with the wall.

1. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When the spheres collide, the line joining their centres is parallel to \(\mathbf { j }\), as shown in Figure 1. The coefficient of restitution between the spheres is \(\frac { 3 } { 7 }\). Find, in terms of \(m\), the total kinetic energy lost in the collision.

2. \begin{figure}[h] \end{figure} Figure 2 represents part of the smooth rectangular floor of a sports hall. A ball is at \(B\), 4 m from one wall of the hall and 5 m from an adjacent wall. These two walls are smooth and meet at the corner \(C\). The ball is kicked so that it travels along the floor, bounces off the first wall at the point \(X\) and hits the second wall at the point \(Y\). The point \(Y\) is 7.5 m from the corner \(C\).
The coefficient of restitution between the ball and the first wall is \(\frac { 3 } { 4 }\).
Modelling the ball as a particle, find the distance \(C X\).

A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).

1. Find, in terms of \(V\), the speed of \(S\)
   1. immediately before the collision,
   2. immediately after the collision.
2. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
   

2. \begin{figure}[h] \end{figure} A smooth fixed plane is inclined at an angle \(\alpha\) to the horizontal. A smooth ball \(B\) falls vertically and hits the plane. Immediately before the impact the speed of \(B\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Immediately after the impact the direction of motion of \(B\) is horizontal. The coefficient of restitution between \(B\) and the plane is \(\frac { 1 } { 3 }\). Find the size of angle \(\alpha\).

1. A smooth uniform sphere \(A\), of mass \(5 m\) and radius \(r\), is at rest on a smooth horizontal plane. A second smooth uniform sphere \(B\), of mass \(3 m\) and radius \(r\), is moving in a straight line on the plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes \(A\). Immediately before the impact the direction of motion of \(B\) makes an angle of \(60 ^ { \circ }\) with the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(30 ^ { \circ }\) by the impact.

Find

1. the speed of \(B\) immediately after the impact,
2. the coefficient of restitution between the spheres.
   

3. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) with equal radii have masses \(m\) and \(2 m\) respectively. The spheres are moving in opposite directions on a smooth horizontal surface and collide obliquely. Immediately before the collision, \(A\) has speed \(3 u\) with its direction of motion at an angle \(\theta\) to the line of centres, and \(B\) has speed \(u\) with its direction of motion at an angle \(\theta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the spheres is \(\frac { 1 } { 8 }\) Immediately after the collision, the speed of \(A\) is twice the speed of \(B\).
Find the size of the angle \(\theta\).

7. \begin{figure}[h] \end{figure} Figure 4 represents the plan view of part of a smooth horizontal floor, where \(A B\) and \(B C\) are smooth vertical walls. The angle between \(A B\) and \(B C\) is \(120 ^ { \circ }\). A ball is projected along the floor towards \(A B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a path at an angle of \(60 ^ { \circ }\) to \(A B\). The ball hits \(A B\) and then hits \(B C\). The ball is modelled as a particle. The coefficient of restitution between the ball and each wall is \(\frac { 1 } { 2 }\)

1. Show that the speed of the ball immediately after it has hit \(A B\) is \(\frac { \sqrt { 7 } } { 4 } u\). The speed of the ball immediately after it has hit \(B C\) is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find \(w\) in terms of \(u\).

1. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a second smooth uniform sphere \(B\), which is at rest on the plane. The sphere \(B\) has mass \(4 m\) and the same radius as \(A\). Immediately before the collision the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres of the spheres, as shown in Figure 1. The direction of motion of \(A\) is turned through an angle of \(90 ^ { \circ }\) by the collision and the coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\) Find the value of \(\tan \alpha\).
1.

2. Figure 2 A small spherical ball \(P\) is at rest at the point \(A\) on a smooth horizontal floor. The ball is struck and travels along the floor until it hits a fixed smooth vertical wall at the point \(X\). The angle between \(A X\) and this wall is \(\alpha\), where \(\alpha\) is acute. A second fixed smooth vertical wall is perpendicular to the first wall and meets it in a vertical line through the point \(C\) on the floor. The ball rebounds from the first wall and hits the second wall at the point \(Y\). After \(P\) rebounds from the second wall, \(P\) is travelling in a direction parallel to \(X A\), as shown in Figure 2. The coefficient of restitution between the ball and the first wall is \(e\). The coefficient of restitution between the ball and the second wall is ke. Find the value of \(k\).
2. \includegraphics[max width=\textwidth, alt={}, center]{b01b3a41-3ed1-4104-b20d-4cfb845df4a1-03_582_645_118_648}

2. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(3 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When the spheres collide, the line joining their centres is parallel to \(\mathbf { j }\), as shown in Figure 1.
The coefficient of restitution between the two spheres is \(e\).
The kinetic energy of sphere \(B\) immediately after the collision is \(85 \%\) of its kinetic energy immediately before the collision. Find

1. the velocity of each sphere immediately after the collision,
2. the value of \(e\).

4. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] A line of greatest slope of a fixed smooth plane is parallel to the vector \(( - 4 \mathbf { i } - 3 \mathbf { j } )\). A particle \(P\) falls vertically and strikes the plane. Immediately before the impact, \(P\) has velocity \(- 7 \mathbf { j } \mathrm {~ms} ^ { - 1 }\). Immediately after the impact, \(P\) has velocity \(( - a \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(a\) is a positive constant.

1. Show that \(a = 6\)
2. Find the coefficient of restitution between \(P\) and the plane.

2. A small ball \(B\), moving on a smooth horizontal plane, collides with a fixed smooth vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(\alpha\). The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 4 }\). The kinetic energy of \(B\) immediately after the collision is \(60 \%\) of its kinetic energy immediately before the collision. Find, in degrees, the size of angle \(\alpha\).

7. Two smooth uniform spheres \(A\) and \(B\), of mass 2 kg and 3 kg respectively, and of equal radius, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that, at the instant when \(A\) and \(B\) collide, their line of centres is parallel to \(- \mathbf { i } + \mathbf { j }\).
2. Find the velocity of \(B\) immediately after the collision.
3. Find the coefficient of restitution between \(A\) and \(B\).

1. A smooth sphere \(S\) is moving on a smooth horizontal plane with speed \(u\) when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of \(S\) makes an angle of \(30 ^ { \circ }\) with the wall. The coefficient of restitution between \(S\) and the wall is \(\frac { 1 } { 3 }\).

Find the speed of \(S\) immediately after the collision.

3. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\), moving on a smooth horizontal table, collides with a second identical sphere \(B\) which is at rest on the table. When the spheres collide the line joining their centres makes an angle of \(30 ^ { \circ }\) with the direction of motion of \(A\), as shown in Fig. 1. The coefficient of restitution between the spheres is \(e\). The direction of motion of \(A\) is deflected through an angle \(\theta\) by the collision. Show that \(\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }\).
(10 marks)

4

1. A pulley consists of a central cylinder of wood and an outer ring of steel. The density of the wood is \(700 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) and the density of the steel is \(7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). The pulley has a radius of 20 cm and is 10 cm thick (see Fig. 4.1). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-4_359_661_404_742} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} Find the radius that the central cylinder must have in order that the moment of inertia of the pulley about the axis of symmetry shown in Fig. 4.1 is \(1.5 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
2. Two blocks P and Q of masses 10 kg and 20 kg are connected by a light inextensible string. The string passes over a heavy rough pulley of radius 25 cm . The pulley can rotate freely and the string does not slip. Block P is held at rest in smooth contact with a plane inclined at \(30 ^ { \circ }\) to the horizontal, and block Q is at rest below the pulley (see Fig. 4.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-5_341_917_438_541} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} At \(t \mathrm {~s}\) after the system is released from rest, the pulley has angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) and block P has constant acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up the slope.
   1. Show that the net loss of energy of the two blocks in the first \(t\) seconds of motion is \(87 t ^ { 2 } \mathrm {~J}\) and use the principle of conservation of energy to show that the moment of inertia of the pulley about its axis of rotation is \(\frac { 87 } { 32 } \mathrm {~kg} \mathrm {~m} ^ { 2 }\). When \(t = 3\) a resistive couple is applied to the pulley. This resistive couple has magnitude \(( 2 \omega + k ) \mathrm { Nm }\), where \(k\) is a constant. The couple on the pulley due to tensions in the sections of string is \(\left( \frac { 147 } { 4 } - \frac { 15 } { 8 } \frac { \mathrm {~d} \omega } { \mathrm {~d} t } \right) \mathrm { Nm }\) in the direction of positive \(\omega\).
   2. Write down a first order differential equation for \(\omega\) when \(t \geqslant 3\) and show by integration that $$\omega = \frac { 1 } { 8 } \left( ( 45 + 4 k ) \mathrm { e } ^ { \frac { 64 } { 147 } ( 3 - t ) } + 147 - 4 k \right) .$$
   3. By considering the equation given in part (ii), find the value or set of values of \(k\) for which the pulley
      (A) continues to rotate with constant angular velocity,
      (B) rotates with decreasing angular velocity without coming to rest,
      (C) rotates with decreasing angular velocity and comes to rest if there is sufficient distance between P and the pulley. \section*{END OF QUESTION PAPER}

4 A solid cylinder of radius \(a \mathrm {~m}\) and length \(3 a \mathrm {~m}\) has density \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\) given by \(\rho = k \left( 2 + \frac { x } { a } \right)\) where \(x \mathrm {~m}\) is the distance from one end and \(k\) is a positive constant. The mass of the cylinder is \(M \mathrm {~kg}\) where \(M = \frac { 21 } { 2 } \pi a ^ { 3 } k\). Let A and B denote the circular faces of the cylinder where \(x = 0\) and \(x = 3 a\), respectively.

1. Show by integration that the moment of inertia, \(I _ { \mathrm { A } } \mathrm { kg } \mathrm { m } ^ { 2 }\), of the cylinder about a diameter of the face A is given by \(I _ { \mathrm { A } } = \frac { 109 } { 28 } M a ^ { 2 }\).
2. Show that the centre of mass of the cylinder is \(\frac { 12 } { 7 } a \mathrm {~m}\) from A .
3. Using the parallel axes theorem, or otherwise, show that the moment of inertia, \(I _ { \mathrm { B } } \mathrm { kg } \mathrm { m } ^ { 2 }\), of the cylinder about a diameter of the face B is given by \(I _ { \mathrm { B } } = \frac { 73 } { 28 } M a ^ { 2 }\). You are now given that \(M = 4\) and \(a = 0.7\). The cylinder is at rest and can rotate freely about a horizontal axis which is a diameter of the face B as shown in Fig. 4. It is struck at the bottom of the curved surface by a small object of mass 0.2 kg which is travelling horizontally at speed \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical plane which is both perpendicular to the axis of rotation and contains the axis of symmetry of the cylinder. The object sticks to the cylinder at the point of impact. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-4_606_435_1087_817} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
4. Find the initial angular speed of the combined object after the collision. \section*{END OF QUESTION PAPER}

3. A uniform rectangular lamina \(A B C D\), where \(A B = a\) and \(B C = 2 a\), has mass \(2 m\). The lamina is free to rotate about its edge \(A B\), which is fixed and vertical. The lamina is at rest when it is struck at \(C\) by a particle \(P\) of mass \(m\). The particle \(P\) is moving horizontally with speed \(U\) in a direction which is perpendicular to the lamina. The coefficient of restitution between \(P\) and the lamina is 0.5 Find the angular speed of the lamina immediately after the impact.
(8)

5 A small ball is held at a height of 160 cm above a horizontal surface. The ball is released from rest and rebounds from the surface. After its first bounce on the surface the ball reaches a height of 122.5 cm .

1. Find the height reached by the ball after its second bounce on the surface. After \(n\) bounces the height reached by the ball is less than 10 cm .
2. Find the minimum possible value of \(n\).
3. State what would happen if the same ball is released from rest from a height of 160 cm above a different horizontal surface and
   (A) the coefficient of restitution between the ball and the new surface is 0 ,
   (B) the coefficient of restitution between the ball and the new surface is 1 .