6.03k Newton's experimental law: direct impact

1. Two particles, \(P\) and \(Q\), of masses \(3 m\) and \(2 m\) respectively, are moving on a smooth horizontal plane. They are moving in opposite directions along the same straight line when they collide directly.

Immediately before the collision, \(P\) is moving with speed \(2 u\).
The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is \(\frac { 9 m u } { 2 }\)

1. Find the speed of \(P\) immediately after the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
   Given that the speed of \(Q\) immediately before the collision is \(u\),
2. find the value of \(e\).

4. \begin{figure}[h] \end{figure} Three particles, \(P , Q\) and \(R\), lie at rest on a smooth horizontal plane. The particles are in a straight line with \(Q\) between \(P\) and \(R\), as shown in Figure 1 . Particle \(P\) is projected towards \(Q\) with speed \(u\). At the same time, \(R\) is projected with speed \(\frac { 1 } { 2 } u\) away from \(Q\), in the direction \(Q R\). Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(2 m\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision between \(P\) and \(Q\) is $$\frac { u ( 1 + e ) } { 3 }$$ It is given that \(e > \frac { 1 } { 2 }\)
2. Determine whether there is a collision between \(Q\) and \(R\).
3. Determine the direction of motion of \(P\) immediately after the collision between \(P\) and \(Q\).
4. Find, in terms of \(m , u\) and \(e\), the total kinetic energy lost in the collision between \(P\) and \(Q\), simplifying your answer.
5. Explain how using \(e = 1\) could be used to check your answer to part (d).

1. A particle \(A\) has mass \(2 m\) and a particle \(B\) has mass \(3 m\). The particles are moving in opposite directions along the same straight line and collide directly.

Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(0.5 u\) and the speed of \(B\) is \(w\). Given that the direction of motion of each particle is reversed by the collision,

1. find \(w\) in terms of \(u\)
2. find the coefficient of restitution between the particles,
3. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(A\) in the collision.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) and a particle \(Q\) of mass \(4 m\) are at rest on a smooth horizontal plane, as shown in Figure 2. Particle \(P\) is projected with speed \(u\) along the plane towards \(Q\) and the particles collide.
The coefficient of restitution between the particles is \(e\), where \(e > \frac { 1 } { 4 }\) As a result of the collision, the direction of motion of \(P\) is reversed and \(P\) has speed \(\frac { u } { 5 } ( 4 e - 1 )\).

1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. After the collision, \(P\) goes on to hit a vertical wall which is fixed at right angles to the direction of motion of \(P\). The coefficient of restitution between \(P\) and the wall is \(f\), where \(f > 0\) Given that \(e = \frac { 3 } { 4 }\)
2. find, in terms of \(m , u\) and \(f\), the kinetic energy lost by \(P\) as a result of its impact with the wall. Give your answer in its simplest form. After its impact with the wall, \(P\) goes on to collide with \(Q\) again.
3. Find the complete range of possible values of \(f\).

2. \begin{figure}[h] \end{figure} Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) are fixed vertical walls with \(A B\) perpendicular to \(B C\). A small ball is projected along the floor towards \(A B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) on a path that makes an angle \(\alpha\) with \(A B\), where \(\tan \alpha = \frac { 4 } { 3 }\). The ball hits \(A B\) and then hits \(B C\).
Immediately after hitting \(A B\), the ball is moving at an angle \(\beta\) to \(A B\), where \(\tan \beta = \frac { 1 } { 3 }\) The coefficient of restitution between the ball and \(A B\) is \(e\).
The coefficient of restitution between the ball and \(B C\) is \(\frac { 1 } { 2 }\) By modelling the ball as a particle and the floor and walls as being smooth,

1. show that the value of \(e = \frac { 1 } { 4 }\)
2. find the speed of the ball immediately after it hits \(B C\).
3. Suggest two ways in which the model could be refined to make it more realistic.

1. A particle \(P\) of mass \(3 m\) and a particle \(Q\) of mass \(2 m\) are moving along the same straight line on a smooth horizontal plane. The particles are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2 u\).
Immediately after the collision \(P\) and \(Q\) are moving in opposite directions.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the range of possible values of \(e\), justifying your answer. Given that \(Q\) loses 75\% of its kinetic energy as a result of the collision,
2. find the value of \(e\).

1. Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\).

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

1. Show that the direction of motion of each of the particles is unchanged by the collision.
   (8) After the collision with \(A\), particle \(B\) collides directly with a third particle, \(C\), of mass \(2 m\), which is at rest on the surface. The coefficient of restitution between \(B\) and \(C\) is also \(e\).
2. Show that there will be a second collision between \(A\) and \(B\).

1. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.

Particle \(A\) has mass \(5 m\) and particle \(B\) has mass \(3 m\).
The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\) Immediately after the collision the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\).
Given that \(A\) and \(B\) are moving in the same direction after the collision,

1. find the set of possible values of \(e\). Given also that the kinetic energy of \(A\) immediately after the collision is \(16 \%\) of the kinetic energy of \(A\) immediately before the collision,
2. find
   1. the value of \(e\),
   2. the magnitude of the impulse received by \(A\) in the collision, giving your answer in terms of \(m\) and \(v\).

1. A particle \(A\) of mass \(3 m\) and a particle \(B\) of mass \(m\) are moving along the same straight line on a smooth horizontal surface. The particles are moving in opposite directions towards each other when they collide directly.

Immediately before the collision, the speed of \(A\) is \(k u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\). The magnitude of the impulse received by \(B\) in the collision is \(\frac { 3 } { 2 } m u\).

1. Find \(v\) in terms of \(u\) only.
2. Find the two possible values of \(k\).

4. \begin{figure}[h] \end{figure} Two smooth uniform spheres, \(A\) and \(B\), have equal radii. The mass of \(A\) is \(3 m\) and the mass of \(B\) is \(4 m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, \(A\) is moving with speed \(3 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres and \(B\) is moving with speed \(2 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(90 ^ { \circ }\) by the collision, as shown in Figure 3.

1. Find the size of the angle through which the direction of motion of \(A\) is turned as a result of the collision.
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision.

8. \begin{figure}[h] \end{figure} Figure 5 represents the plan view of part of a smooth horizontal floor, where \(R S\) and \(S T\) are smooth fixed vertical walls. The vector \(\overrightarrow { R S }\) is in the direction of \(\mathbf { i }\) and the vector \(\overrightarrow { S T }\) is in the direction of \(( 2 \mathbf { i } + \mathbf { j } )\). A small ball \(B\) is projected across the floor towards \(R S\). Immediately before the impact with \(R S\), the velocity of \(B\) is \(( 6 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The ball bounces off \(R S\) and then hits \(S T\). The ball is modelled as a particle.
Given that the coefficient of restitution between \(B\) and \(R S\) is \(e\),

1. find the full range of possible values of \(e\). It is now given that \(e = \frac { 1 } { 4 }\) and that the coefficient of restitution between \(B\) and \(S T\) is \(\frac { 1 } { 2 }\)
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after its impact with \(S T\).

5. \begin{figure}[h] \end{figure} A smooth uniform sphere \(S\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(S\) collides obliquely with another uniform sphere of mass \(M\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision the direction of motion of \(S\) makes an angle \(\alpha\), where \(0 < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(S\) makes an angle \(\beta\) with the line joining the centres of the spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that \(\tan \beta = \frac { ( m + M ) \tan \alpha } { ( m - e M ) }\) Given that \(m = e M\),
2. show that the directions of motion of the two spheres immediately after the collision are perpendicular.

1. A particle \(P\) of mass \(m\) is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(0 < \alpha \leqslant 45 ^ { \circ }\)

At the instant immediately before the impact, the speed of \(P\) is \(u\).
At the instant immediately after the impact, \(P\) is moving horizontally with speed \(v\).

1. Show that the magnitude of the impulse exerted on the plane by \(P\) is \(m u \sec \alpha\) The coefficient of restitution between \(P\) and the plane is \(e\), where \(e > 0\)
2. Show that \(v ^ { 2 } = u ^ { 2 } \left( \sin ^ { 2 } \alpha + e ^ { 2 } \cos ^ { 2 } \alpha \right)\)
3. Show that the kinetic energy lost by \(P\) in the impact is $$\frac { 1 } { 2 } m u ^ { 2 } \left( 1 - e ^ { 2 } \right) \cos ^ { 2 } \alpha$$
4. Hence find, in terms of \(m\), \(u\) and \(e\) only, the kinetic energy lost by \(P\) in the impact.

7. \begin{figure}[h] \end{figure} A small smooth snooker ball is projected from the corner \(A\) of a horizontal rectangular snooker table \(A B C D\). The ball is projected so it first hits the side \(D C\) at the point \(P\), then hits the side \(C B\) at the point \(Q\) and then returns to \(A\). Angle \(A P D = \alpha\), Angle \(Q P C = \beta\), Angle \(A Q B = \gamma\) The ball moves along \(A P\) with speed \(U\), along \(P Q\) with speed \(V\) and along \(Q A\) with speed \(W\), as shown in Figure 2. The coefficient of restitution between the ball and side \(D C\) is \(e _ { 1 }\) The coefficient of restitution between the ball and side \(C B\) is \(e _ { 2 }\) The ball is modelled as a particle. \section*{Use the model to answer all parts of this question.}

1. Show that \(\tan \beta = e _ { 1 } \tan \alpha\)
2. Hence show that \(e _ { 1 } \tan \alpha = e _ { 2 } \cot \gamma\)
3. By considering (angle \(A P Q\) + angle \(A Q P\) ) or otherwise, show that it would be possible for the ball to return to \(A\) only if \(e _ { 2 } > e _ { 1 }\) If instead \(e _ { 1 } = e _ { 2 }\), the ball would not return to \(A\).
   Given that \(e _ { 1 } = e _ { 2 }\)
4. use the result from part (b) to describe the path of the ball after it hits \(C B\) at \(Q\), explaining your answer.

1. A particle \(A\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. Particle \(A\) collides directly with a particle \(B\) of mass \(m\) which is at rest on the plane.

The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\)

1. Show that the speed of \(B\) immediately after the collision is \(2 u ( 1 + e )\). After the collision, \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\).
2. Show that there will be a second collision between \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\) Find, in simplified form, in terms of \(m\), \(u\) and \(e\),
3. the magnitude of the impulse received by \(B\) in its collision with the wall,
4. the loss in kinetic energy of \(B\) due to its collision with the wall.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]

A particle \(P\) is moving with velocity ( \(4 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) on a smooth horizontal plane. The particle collides with a smooth vertical wall and rebounds with velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The coefficient of restitution between \(P\) and the wall is \(e\).

1. Find the value of \(e\). After the collision, \(P\) goes on to hit a second smooth vertical wall, which is parallel to \(\mathbf { i }\).
   The coefficient of restitution between \(P\) and this second wall is \(\frac { 1 } { 3 }\) The angle through which the direction of motion of \(P\) has been deflected by its collision with this second wall is \(\alpha ^ { \circ }\).
2. Find the value of \(\alpha\), giving your answer to the nearest whole number.

7. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(A\) collides obliquely with a smooth uniform sphere \(B\) of mass \(3 m\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\), where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision, the direction of motion of \(A\) is perpendicular to its original direction, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) immediately after the collision is $$\frac { 1 } { 4 } ( 1 + e ) U \cos \alpha$$
2. Show that \(e > \frac { 1 } { 3 }\)
3. Show that \(0 < \tan \alpha \leqslant \frac { 1 } { \sqrt { 2 } }\)

1. A particle of mass \(m \mathrm {~kg}\) lies on a smooth horizontal surface.

Initially the particle is at rest at a point \(O\) between two fixed parallel vertical walls.
The point \(O\) is equidistant from the two walls and the walls are 4 m apart.
At time \(t = 0\) the particle is projected from \(O\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls.
The coefficient of restitution between the particle and each wall is \(\frac { 3 } { 4 }\) The magnitude of the impulse on the particle due to the first impact with a wall is \(\lambda m u\) Ns.

1. Find the value of \(\lambda\). The particle returns to \(O\), having bounced off each wall once, at time \(t = 7\) seconds.
2. Find the value of \(u\).

4. \begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) are perpendicular vertical walls. The floor and the walls are modelled as smooth.
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 wall \(A B\) is \(\frac { 1 } { \sqrt { 3 } }\) The coefficient of restitution between the ball and wall \(B C\) is \(\sqrt { \frac { 2 } { 5 } }\)

1. Show that, using this model, the final kinetic energy of the ball is \(35 \%\) of the initial kinetic energy of the ball.
2. In reality the floor and the walls may not be smooth. What effect will the model have had on the calculation of the percentage of kinetic energy remaining?

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere \(A\) has mass \(2 m \mathrm {~kg}\) and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(3 m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane when they collide obliquely.
Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The coefficient of restitution between the spheres is \(\frac { 1 } { 4 }\)

1. Find the velocity of \(B\) immediately after the collision.
2. Find, to the nearest degree, the size of the angle through which the direction of motion of \(B\) is deflected as a result of the collision.