6.03l Newton's law: oblique impacts

\includegraphics{figure_4} Two identical smooth uniform spheres \(A\) and \(B\) each have mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(2u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(30°\) with the line of centres, and \(B\)'s direction of motion is perpendicular to the line of centres (see diagram). After the collision, \(A\) and \(B\) are moving in the same direction. The coefficient of restitution between the spheres is \(e\).

1. Find the value of \(e\). [5]
2. Find the loss in the total kinetic energy of the spheres as a result of the collision. [3]

Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). \includegraphics{figure_1} Find the value of \(\tan\theta\). [6]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). Find the value of \(\tan\theta\). [6]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(\frac{1}{2}u\) respectively. Immediately before the collision, \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). As a result of the collision, the direction of motion of \(A\) is reversed and its speed is reduced to \(\frac{1}{4}u\). The direction of motion of \(B\) again makes an angle \(\theta\) with the line of centres, but on the opposite side of the line of centres. The speed of \(B\) is unchanged. Find the value of the coefficient of restitution between the spheres. [4]

Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is moving on the floor with velocity \(u\) and collides directly with \(B\). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) after the collision. [3]

Subsequently, \(B\) collides with a fixed vertical wall which makes an angle \(\theta\) with the direction of motion of \(B\), where \(\tan\theta = \frac{3}{4}\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{3}\). Immediately after \(B\) collides with the wall, the kinetic energy of \(B\) is \(\frac{5}{27}\) of the kinetic energy of \(B\).

1. Find the possible values of \(e\). [7]

\includegraphics{figure_7} The smooth vertical walls \(AB\) and \(CB\) are at right angles to each other. A particle \(P\) is moving with speed \(u\) on a smooth horizontal floor and strikes the wall \(CB\) at an angle \(\alpha\). It rebounds at an angle \(\beta\) to the wall \(CB\). The particle then strikes the wall \(AB\) and rebounds at an angle \(\gamma\) to that wall (see diagram). The coefficient of restitution between each wall and \(P\) is \(e\).

1. Show that \(\tan \beta = e \tan \alpha\). [3]
2. Express \(\gamma\) in terms of \(\alpha\) and explain what this result means about the final direction of motion of \(P\). [4]

As a result of the two impacts the particle loses \(\frac{8}{9}\) of its initial kinetic energy.

1. Given that \(\alpha + \beta = 90°\), find the value of \(e\) and the value of \(\tan \alpha\). [4]

\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{2}{3}m\) respectively. The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle of \(60°\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{2}{3}\).

1. Find the angle through which the direction of motion of \(B\) is deflected by the collision. [6]
2. Find the loss in the total kinetic energy of the system as a result of the collision. [3]

\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 \(\frac{5}{8}\) 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]

\includegraphics{figure_1} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving with equal speeds \(u\) on a smooth horizontal surface when they collide. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(60°\) with the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(e\). After the collision, the component of the velocity of \(A\) along the line of centres is \(v\) and \(B\) moves perpendicular to the line of centres. Sphere \(A\) now has twice as much kinetic energy as sphere \(B\).

1. Show that \(v = \frac{1}{2}u(4\cos\theta - 1)\). [1]
2. Find the value of \(\cos\theta\). [4]
3. Find the value of \(e\). [2]

\includegraphics{figure_4} Two smooth vertical walls meet at right angles. The smooth sphere \(A\), with mass \(m\), is at rest on a smooth horizontal surface and is at a distance \(d\) from each wall. An identical smooth sphere \(B\) is moving on the horizontal surface with speed \(u\) at an angle \(\theta\) with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Find the value of \(\tan \theta\). [4]
2. Find the percentage loss in the total kinetic energy of the spheres as a result of this collision. [3]

\includegraphics{figure_3} The diagram shows two identical smooth uniform spheres \(A\) and \(B\) of equal radii and each of mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(2u\) and \(3u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle \(\theta\) with the line of centres and \(B\)'s direction of motion is perpendicular to that of \(A\). After the collision, \(B\) moves perpendicular to the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{3}\).

1. Find the value of \(\tan \theta\). [3]
2. Find the total loss of kinetic energy as a result of the collision. [2]
3. Find, in degrees, the angle through which the direction of motion of \(A\) is deflected as a result of the collision. [2]

A smooth sphere \(P\) of mass \(m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(2m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). After the collision the direction of motion of \(P\) is unchanged. The spheres have the same radii and the coefficient of restitution between \(P\) and \(Q\) is \(e\). By modelling the spheres as particles,

1. show that the speed of \(Q\) immediately after the collision is \(\frac{1}{3}(1 + e)u\), [5]

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

Given that \(e = \frac{1}{4}\),

1. find the loss of kinetic energy in the collision. [4]

1. Give one possible form of energy into which the lost kinetic energy has been transformed. [1]

TURN OVER FOR QUESTION 7

A particle is projected from a point with speed \(u\) at an angle of elevation \(α\) above the horizontal and moves freely under gravity. When it has moved a horizontal distance \(x\), its height above the point of projection is \(y\).

1. Show that $$y = x \tan α - \frac{gx^2}{2u^2}(1 + \tan^2 α).$$ [5]

A shot-putter puts a shot from a point \(A\) at a height of 2 m above horizontal ground. The shot is projected at an angle of elevation of \(45°\) with a speed of 14 m s\(^{-1}\). By modelling the shot as a particle moving freely under gravity,

1. find, to 3 significant figures, the horizontal distance of the shot from \(A\) when the shot hits the ground, [5]

1. find, to 2 significant figures, the time taken by the shot in moving from \(A\) to reach the ground. [2]

A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(v\) when it collides directly with another small smooth ball \(B\) of mass \(3m\) which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is \(e\). The direction of motion of \(A\) is reversed as a result of the collision.

1. Find, in terms of \(e\) and \(u\), the speeds of \(A\) and \(B\) immediately after the collision. [7]

In the subsequent motion \(B\) strikes a vertical wall, which is perpendicular to the direction of motion of \(B\), and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac{1}{2}\). Given that there is a second collision between \(A\) and \(B\),

1. find the range of values of \(e\) for which the motion described is possible. [6]

\includegraphics{figure_2} A smooth sphere \(P\) of mass \(2m\) is moving in a straight line with speed \(v\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). The spheres are modelled as particles.

1. Show that, immediately after the collision, the speeds of \(P\) and \(Q\) are \(\frac{3}{4}u\) and \(\frac{5}{4}u\) respectively. [7]

After the collision, \(Q\) strikes a fixed vertical wall which is perpendicular to the direction of motion of \(P\) and \(Q\). The coefficient of restitution between \(Q\) and the wall is \(e\). When \(P\) and \(Q\) collide again, \(P\) is brought to rest.

1. Find the value of \(e\). [7]

1. Explain why there must be a third collision between \(P\) and \(Q\). [1]

Show that \(GX = \frac{44}{63}a\). [6] The mass of the lamina is \(M\). A particle of mass \(λM\) is attached to the lamina at \(C\). The lamina is suspended from \(B\) and hangs freely under gravity with \(AB\) horizontal.

1. Find the value of \(λ\). [3]

TURN OVER FOR QUESTION 7

A smooth sphere \(P\) of mass \(m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(2m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). After the collision the direction of motion of \(P\) is unchanged. The spheres have the same radii and the coefficient of restitution between \(P\) and \(Q\) is \(e\). By modelling the spheres as particles,

1. show that the speed of \(Q\) immediately after the collision is \(\frac{1}{3}(1 + e)u\), [5]
2. find the range of possible values of \(e\). [4]

Given that \(e = \frac{1}{4}\),

1. find the loss of kinetic energy in the collision. [4]
2. Give one possible form of energy into which the lost kinetic energy has been transformed. [1]

A smooth sphere \(P\) of mass \(2m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). The spheres are modelled as particles.

1. Show that, immediately after the collision, the speeds of \(P\) and \(Q\) are \(\frac{2}{9}u\) and \(\frac{8}{9}u\) respectively. [7]

After the collision, \(Q\) strikes a fixed vertical wall which is perpendicular to the direction of motion of \(P\) and \(Q\). The coefficient of restitution between \(Q\) and the wall is \(e\). When \(P\) and \(Q\) collide again, \(P\) is brought to rest.

1. Find the value of \(e\). [7]
2. Explain why there must be a third collision between \(P\) and \(Q\). [1]

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]