6.03k Newton's experimental law: direct impact

\includegraphics{figure_4} Two smooth particles \(P\) and \(Q\) have masses \(m\) and \(2m\) respectively. The particles are moving in the same direction in the same straight line, on a smooth horizontal plane, with \(Q\) in front of \(P\). The particles are moving towards a fixed smooth vertical wall which is perpendicular to the direction of motion of the particles, as shown in Figure 4. The speed of \(P\) is \(2u\) and the speed of \(Q\) is \(3u\). The coefficient of restitution between \(Q\) and the wall is \(\frac{1}{3}\). Particle \(Q\) strikes the wall, rebounds and then collides directly with \(P\). The direction of motion of each particle is reversed by this collision. Immediately after this collision the speed of \(P\) is \(v\) and the speed of \(Q\) is \(w\).

1. Show that \(v = 2w\). [5]

The total kinetic energy of \(P\) and \(Q\) immediately after they collide is half the total kinetic energy of \(P\) and \(Q\) immediately before they collide.

1. Find the coefficient of restitution between \(P\) and \(Q\). [8]

Two smooth spheres \(X\) and \(Y\), of masses \(x\) kg and \(y\) kg respectively, are free to move in a smooth straight groove in a horizontal table. \(X\) is projected with speed \(6\) ms\(^{-1}\) towards \(Y\), which is stationary. After the collision \(X\) moves with speed \(2\) ms\(^{-1}\) and \(Y\) moves with speed \(3\) ms\(^{-1}\).

1. Calculate the two possible values of the ratio \(x : y\). \hfill [6 marks]
2. State a modelling assumption that you have made concerning \(X\) and \(Y\). \hfill [1 mark]

\(Y\) now strikes a vertical barrier and rebounds along the groove with speed \(k\) ms\(^{-1}\), colliding again with \(X\) which is still moving at \(2\) ms\(^{-1}\). Given that in this impact \(Y\) is brought to rest and the direction of motion of \(X\) is reversed,

1. show that \(k > 1.5\). \hfill [4 marks]

Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. \(P\) has speed 4 ms\(^{-1}\). They collide directly. After the collision the direction of motion of both particles has been reversed, and \(Q\) has speed 2 ms\(^{-1}\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). Find

1. the speed of \(Q\) before the collision, [4 marks]
2. the speed of \(P\) after the collision, [4 marks]
3. the kinetic energy, in J, lost in the impact. [4 marks]

Two railway trucks, \(P\) and \(Q\), of equal mass, are moving towards each other with speeds \(4u\) and \(5u\) respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. [6 marks]
2. Show that \(e > \frac{1}{9}\). [2 marks]

\(Q\) now hits a fixed buffer and rebounds along the track. \(P\) continues to move with the speed that it had immediately after it collided with \(Q\).

1. Prove that it is impossible for a further collision between \(P\) and \(Q\) to occur. [3 marks]

Three particles \(A\), \(B\) and \(C\), of equal size and each of mass \(m\), are at rest on the same straight line on a smooth horizontal surface. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(e\). \(A\) is projected with speed \(7\) ms\(^{-1}\) and strikes \(B\) directly. \(B\) then collides with \(C\), which starts to move with speed \(4\) ms\(^{-1}\). Calculate the value of \(e\). [10 marks]

Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3m\) and \(4m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.

1. Show that \(B\) begins to move with speed \(\frac{3}{7}u(1 + e)\). [6 marks]
2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). [3 marks]

Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(kmu\) Ns, where \(k > 2.25\). \(A\) then collides again with \(B\).

1. Show that the speed of \(A\) after this second impact is independent of \(k\). [7 marks]

Two railway trucks \(A\) and \(B\), whose masses are \(6m\) and \(5m\) respectively, are moving in the same direction along a straight track with speeds \(5u\) and \(3u\) respectively, and collide directly. Immediately after this impact the speeds of \(A\) and \(B\) are \(v\) and \(kv\) respectively, in the same direction as before. The coefficient of restitution between \(A\) and \(B\) is \(e\). Modelling the trucks as particles,

1. show that
   1. \(v = \frac{45u}{5k + 6}\),
   2. \(v = \frac{2eu}{k - 1}\).
   [8 marks]
2. Use the fact that \(0 \leq e \leq 1\) to deduce the range of possible values of \(k\). [5 marks]

A heavy ball, of mass 2 kg, rolls along a horizontal surface. It strikes a vertical wall at a speed of 4 ms\(^{-1}\) and rebounds. The coefficient of restitution between the ball and the wall is 0.4. Find the kinetic energy lost in the impact. [5 marks]

Two smooth spheres \(A\) and \(B\), of equal radius and masses \(9m\) and \(4m\) respectively, are moving towards each other along a straight line with speeds 4 ms\(^{-1}\) and 6 ms\(^{-1}\) respectively. They collide, after which the direction of motion of \(A\) remains unchanged.

1. Show that the speed of \(B\) after the impact cannot be more than 3 ms\(^{-1}\). [5 marks]

The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that \(e < \frac{3}{10}\). [5 marks]
2. Find the speeds of \(A\) and \(B\) after the impact in the case when \(e = 0\). [4 marks]

\includegraphics{figure_4} Three smooth spheres \(A\), \(B\) and \(C\), of equal radius and of masses \(m\) kg, \(2m\) kg and \(3m\) kg respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed 5 m s\(^{-1}\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed 2 m s\(^{-1}\).

1. Find the coefficient of restitution between \(A\) and \(B\). [4]
2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. [2]

Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.

1. Show that there will be another collision. [3]

A particle \(A\) is released from rest from the top of a smooth plane, which makes an angle of 30° with the horizontal. The particle \(A\) collides 2 s later with a particle \(B\), which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of \(B\) immediately before the collision is 2 ms\(^{-1}\). \(B\) has velocity 1 ms\(^{-1}\) down the plane immediately after the collision. Find

1. the speed of \(A\) immediately after the collision, [4]
2. the distance \(A\) moves up the plane after the collision. [2]

The masses of \(A\) and \(B\) are 0.5 kg and \(m\) kg, respectively.

1. Find the value of \(m\). [3]

A particle \(A\) of mass \(2m\) is moving with speed \(u\) on a smooth horizontal surface when it collides with a stationary particle \(B\) of mass \(m\). After the collision the speed of \(A\) is \(v\), the speed of \(B\) is \(3v\) and the particles move in the same direction.

1. Find \(v\) in terms of \(u\). [3]
2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{3}\). [2]

\(B\) subsequently hits a vertical wall which is perpendicular to the direction of motion. As a result of the impact, \(B\) loses \(\frac{3}{4}\) of its kinetic energy.

1. Show that the speed of \(B\) after hitting the wall is \(\frac{3}{4}u\). [4]
2. \(B\) then hits \(A\). Calculate the speeds of \(A\) and \(B\), in terms of \(u\), after this collision and state their directions of motion. [8]

The masses of two particles \(A\) and \(B\) are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(8 \text{ m s}^{-1}\) and \(B\) has speed \(10 \text{ m s}^{-1}\) before they collide. The kinetic energy lost due to the collision is 121.5 J.

1. Find the speed and direction of motion of each particle after the collision. [8]
2. Find the coefficient of restitution between \(A\) and \(B\). [2]

A particle \(P\) is projected with speed \(32 \text{ m s}^{-1}\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac{3}{4}\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac{24}{25}\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).

1. Calculate the height of \(C\) above the ground and the distance \(AB\). [4]

Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.

1. Given that the mass of \(P\) is 3 kg, find the magnitude and direction of the impulse exerted on \(P\) by the ground. [4]

The coefficient of restitution between the two particles is \(\frac{1}{2}\).

1. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25°\) below the horizontal. [9]

A sledge and a child sitting on it have a combined mass of 29.5 kg. The sledge slides on horizontal ice with negligible resistance to its movement.

1. While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The coefficient of restitution in the collision is 0.8. After the impact the speeds of the sledge and the ball are \(V_1 \text{ m s}^{-1}\) and \(V_2 \text{ m s}^{-1}\) respectively. Calculate \(V_1\) and \(V_2\) and state the direction in which the ball is travelling after the impact. [7]
2. While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The snowball sticks to the sledge.
   1. Calculate the velocity with which the combined sledge and snowball start to move. [3]
   2. The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer. [2]
3. In another situation, the sledge is travelling over the ice at \(2 \text{ m s}^{-1}\) with 10.5 kg of snow on it (giving a total mass of 40 kg). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \text{ m s}^{-1}\) and the snowball and sledge are separating at a speed of \(10 \text{ m s}^{-1}\). Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown. Calculate \(V\). [5]

Fig. 1.1 shows block A of mass 2.5 kg which has been placed on a long, uniformly rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.8\). The coefficient of friction between A and the slope is 0.85. \includegraphics{figure_1}

1. Calculate the maximum possible frictional force between A and the slope. Show that A will remain at rest. [6]

With A still at rest, block B of mass 1.5 kg is projected down the slope, as shown in Fig. 1.2. B has a speed of 16 m s\(^{-1}\) when it collides with A. In this collision the coefficient of restitution is 0.4, the impulses are parallel to the slope and linear momentum parallel to the slope is conserved.

1. Show that the velocity of A immediately after the collision is 8.4 m s\(^{-1}\) down the slope. Find the velocity of B immediately after the collision. [6]
2. Calculate the impulse on B in the collision. [3]

The blocks do not collide again.

1. For what length of time after the collision does A slide before it comes to rest? [4]

A smooth uniform sphere \(A\), of mass \(m\), is moving with velocity \(8u\) in a straight line on a smooth horizontal table. A smooth uniform sphere \(B\), of mass \(4m\), has the same radius as \(A\) and is moving on the table with velocity \(u\). \includegraphics{figure_4} The sphere \(A\) collides directly with the sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. 1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) immediately after the collision. [6 marks]
   2. The direction of motion of \(A\) is reversed by the collision. Show that \(e > a\), where \(a\) is a constant to be determined. [2 marks]
2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{5}\). The sphere \(B\) collides with \(A\) again after rebounding from the wall. Show that \(e < b\), where \(b\) is a constant to be determined. [3 marks]
3. Given that \(e = \frac{4}{7}\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by the wall. [3 marks]

A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20°\) to the horizontal. The point \(O\) is at a perpendicular distance of \(1\) m from the inclined plane. The ball is projected with velocity \(22 \text{ m s}^{-1}\) at an angle of \(70°\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\). \includegraphics{figure_5}

1. 1. Find the time taken by the ball to travel from \(O\) to \(A\). [4 marks]
   2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\). [4 marks]
2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\). Show that \(e < k\), where \(k\) is a constant to be determined. [4 marks]

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]

A small stone \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus \(3mg\) N and natural length \(2l\) m. The other end of the string is fixed to a point \(O\) at a height \(3l\) m above a horizontal surface. \(P\) is released from rest at \(O\); it hits the surface and rebounds to a height of \(2l\) m. The coefficient of restitution between \(P\) and the surface is \(e\). Calculate the value of \(e\). [9 marks] State one assumption (other than the string being light) that you have used in your solution. [1 mark]

A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \text{ m s}^{-1}\) and \(6 \text{ m s}^{-1}\) respectively. The speed of the sphere immediately after it bounces is \(5 \text{ m s}^{-1}\).

1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \text{ m s}^{-1}\), and hence find the coefficient of restitution between the surface and the sphere. [3]
2. State the direction of the impulse on the sphere and find its magnitude. [3]

\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]

A uniform smooth sphere \(A\) moves on a smooth horizontal surface towards a smooth vertical wall. Immediately before the sphere hits the wall it has components of velocity parallel and perpendicular to the wall each of magnitude \(4\) m s\(^{-1}\). Immediately after hitting the wall the components have magnitudes \(u\) m s\(^{-1}\) and \(v\) m s\(^{-1}\), respectively (see Fig. 1). \includegraphics{figure_1}

1. Given that the coefficient of restitution between the sphere and the wall is \(\frac{1}{4}\), state the values of \(u\) and \(v\). [2]

Shortly after hitting the wall the sphere \(A\) comes into contact with another uniform smooth sphere \(B\), which has the same mass and radius as \(A\). The sphere \(B\) is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic. \includegraphics{figure_2}

1. Find, for each sphere, its speed and its direction of motion immediately after the contact. [6]

\includegraphics{figure_7} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length \(0.7\) m. A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(OP\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6\) m s\(^{-1}\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9\) m s\(^{-1}\).

1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\). [3]
2. Given that the speed of \(Q\) is \(v\) m s\(^{-1}\) when \(OQ\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v^2\) in terms of \(\theta\), and show that the tension in the string \(OQ\) is \(14.7m(1 + 2\cos\theta)\) N, where \(m\) kg is the mass of \(Q\). [6]
3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(OQ\) becomes slack. [4]
4. Show that \(V^2 = 0.8575\), where \(V\) m s\(^{-1}\) is the speed of \(Q\) when it reaches its greatest height (after the string \(OQ\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position. [4]