6.03k Newton's experimental law: direct impact

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]

\includegraphics{figure_5} The masses of two spheres \(A\) and \(B\) are \(3m\) kg and \(m\) kg respectively. The spheres are moving towards each other with constant speeds \(2u \, \text{m s}^{-1}\) and \(u \, \text{m s}^{-1}\) respectively along the same straight line towards each other on a smooth horizontal surface (see diagram). The two spheres collide and the coefficient of restitution between the spheres is \(e\). After colliding, \(A\) and \(B\) both move in the same direction with speeds \(v \, \text{m s}^{-1}\) and \(w \, \text{m s}^{-1}\), respectively.

1. Find an expression for \(v\) in terms of \(e\) and \(u\). [6]
2. Write down unsimplified expressions in terms of \(e\) and \(u\) for
   1. the total kinetic energy of the spheres before the collision, [1]
   2. the total kinetic energy of the spheres after the collision. [2]
3. Given that the total kinetic energy of the spheres after the collision is \(\lambda\) times the total kinetic energy before the collision, show that $$\lambda = \frac{27e^2 + 25}{52}.$$ [3]
4. Comment on the cases when
   1. \(\lambda = 1\),
   2. \(\lambda = \frac{25}{52}\). [3]

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]

\includegraphics{figure_6} The rim of a smooth hemispherical bowl is a circle of centre O and radius \(a\). The bowl is fixed with its rim horizontal and uppermost. A particle P of mass \(m\) is released from rest at a point A on the rim as shown in Fig. 6. When P reaches the lowest point of the bowl it collides directly with a stationary particle Q of mass \(\frac{1}{2}m\). After the collision Q just reaches the rim of the bowl. Find the coefficient of restitution between P and Q. [7]

Two uniform smooth spheres A and B have equal radii and are moving on a smooth horizontal surface. The mass of A is 0.2kg and the mass of B is 0.6kg. The spheres collide obliquely. When the spheres collide the line joining their centres is parallel to \(\mathbf{i}\). Immediately before the collision the velocity of A is \(\mathbf{u}_A\)ms\(^{-1}\) and the velocity of B is \(\mathbf{u}_B\)ms\(^{-1}\). The coefficient of restitution between A and B is 0.5. Immediately after the collision the velocity of A is \((-4\mathbf{i} + 2\mathbf{j})\)ms\(^{-1}\) and the velocity of B is \((2\mathbf{i} + 3\mathbf{j})\)ms\(^{-1}\).

1. Find \(\mathbf{u}_A\) and \(\mathbf{u}_B\). [7]

After the collision B collides with a smooth vertical wall which is parallel to \(\mathbf{j}\). The loss in kinetic energy of B caused by the collision with the wall is 1.152J.

1. Find the coefficient of restitution between B and the wall. [3]
2. Find the angle through which the direction of motion of B is deflected as a result of the collision with the wall. [4]

\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]

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]

A particle P of mass 2 kg is projected vertically upwards from horizontal ground with an initial speed of \(14 \text{ m s}^{-1}\). At the same instant a particle Q of mass 8 kg is released from rest 5 m vertically above P. During the subsequent motion P and Q collide. The coefficient of restitution between P and Q is \(\frac{11}{14}\). Determine the time between this collision and P subsequently hitting the ground. [10]

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]

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]

In this question take \(g = 10\). A smooth ball of mass 0.1 kg is projected from a point on smooth horizontal ground with speed 65 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal, where \(\tan\alpha = \frac{3}{4}\). While it is in the air the ball is modelled as a particle moving freely under gravity. The ball bounces on the ground repeatedly. The coefficient of restitution for the first bounce is 0.4.

1. Show that the ball leaves the ground after the first bounce with a horizontal speed of 52 m s\(^{-1}\) and a vertical speed of 15.6 m s\(^{-1}\). Explain your reasoning carefully. [4]
2. Calculate the magnitude of the impulse exerted on the ball by the ground at the first bounce. [2]

Each subsequent bounce is modelled by assuming that the coefficient of restitution is 0.4 and that the bounce takes no time. The ball is in the air for \(T_1\) seconds between projection and bouncing the first time, \(T_2\) seconds between the first and second bounces, and \(T_n\) seconds between the \((n-1)\)th and \(n\)th bounces.

1. 1. Show that \(T_1 = \frac{39}{5}\). [2]
   2. Find an expression for \(T_n\) in terms of \(n\). [2]
2. According to the model, how far does the ball travel horizontally while it is still bouncing? [3]
3. According to the model, what is the motion of the ball after it has stopped bouncing? [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]

Two spheres \(A\) and \(B\), of equal radii, are moving towards each other on a smooth horizontal surface and collide directly. Sphere \(A\) has mass \(4m\) kg and sphere \(B\) has mass \(3m\) kg. Just before the collision, \(A\) has speed \(9\text{ ms}^{-1}\) and \(B\) has speed \(3.5\text{ ms}^{-1}\). Immediately after the collision, \(A\) has speed \(1.5\text{ ms}^{-1}\) in the direction of its original motion.

1. Show that the speed of \(B\) immediately after the collision is \(6.5\text{ ms}^{-1}\). [3]
2. Calculate the coefficient of restitution between \(A\) and \(B\). [3]
3. Given that the magnitude of the impulse exerted by \(B\) on \(A\) is 36 Ns, find the value of \(m\). [3]
4. Give a reason why it is not necessary to model the spheres as particles in this question. [1]

Three particles, \(P\), \(Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(km\) respectively, where \(k\) is a constant. Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly. The coefficient of restitution between each pair of particles is \(e\).

1. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. [9] Given that the mass of \(P\) is \(km\) and that there is a second collision,
2. write down, in terms of \(u\), \(k\) and \(e\), the speed of \(Q\) after this second collision. [1]

\includegraphics{figure_2} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W_1\) and \(W_2\) are two fixed parallel vertical walls. The walls are \(3\) metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leq 3\), from \(W_1\). At time \(t = 0\), the particle is projected from \(O\) towards \(W_1\) with speed \(u\text{ms}^{-1}\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac{2}{3}\). The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.

1. Show that \(T = \frac{45 - 5d}{4u}\). [6]
2. The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer. [2]

\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_5} \(AB\) and \(BC\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(ABC = 60°\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(AB\) at an angle \(\theta\) with \(AB\). It then strikes \(BC\) and rebounds at an angle \(\beta\) with \(BC\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is 40% of its kinetic energy before the first collision.

1. Find the value of \(e\). [4]
2. Find the size of angle \(\beta\). [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]

Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \text{ ms}^{-1}\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics{figure_4}

1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(v_A \text{ ms}^{-1}\) and \(v_B \text{ ms}^{-1}\) respectively. \(\bullet\) Show that \(v_A = \frac{u(3-2e)}{5}\). \(\bullet\) Find an expression for \(v_B\) in terms of \(u\) and \(e\). [4]
2. Find an expression in terms of \(u\) and \(e\) for the velocity of \(B\) immediately after its collision with \(C\). [4]

After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).

1. Determine the range of possible values of \(e\). [4]

A light inextensible string passes over a fixed smooth light pulley. Particles \(A\) and \(B\), of masses 2 kg and 3 kg respectively, are attached to the ends so that the portions of the string below the axis of the pulley are vertical (see diagram). The centre of the horizontal axis of the pulley is 4 m above ground level. \includegraphics{figure_13} The particles are released from rest 1 m above ground level with the string taut.

1. Determine the acceleration of both particles prior to the impact of \(B\) with the ground. [3]
2. Determine the greatest height attained by \(A\) above ground level. [3]
3. If \(B\) rebounds after impact to a first maximum height of 0.05 m, determine the coefficient of restitution between \(B\) and the ground. [2]

\includegraphics{figure_12} A particle \(P\) of mass 2 kg can move along a line of greatest slope on a smooth plane, inclined at \(30°\) to the horizontal. \(P\) is initially at rest at a point on the plane, and a force of constant magnitude 20 N is applied to \(P\) parallel to and up the slope (see diagram).

1. Copy and complete the diagram, showing all forces acting on \(P\). [1]
2. Find the velocity of \(P\) in terms of time \(t\) seconds, whilst the force of 20 N is applied. [4]

After 3 seconds the force is removed at the instant that \(P\) collides with a particle of mass 1 kg moving down the slope with speed 5 m s\(^{-1}\). The coefficient of restitution between the particles is 0.2.

1. Express the velocity of \(P\) as a function of time after the collision. [6]

A ball of mass 0.04 kg is released from rest at a height of 1 metre above a table. It rebounds to a height of 0.81 metre.

1. Find the value of \(e\), the coefficient of restitution. [4]
2. Find the impulse on the ball when it hits the table. [3]

\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]