6.03f Impulse-momentum: relation

1. Three particles \(A , B\) and \(C\) are at rest on a smooth horizontal plane. The particles lie along a straight line with \(B\) between \(A\) and \(C\).

Particle \(B\) has mass \(4 m\) and particle \(C\) has mass \(k m\), where \(k\) is a positive constant. Particle \(B\) is projected with speed \(u\) along the plane towards \(C\) and they collide directly. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 1 } { 4 }\)

1. Find the range of values of \(k\) for which there would be no further collisions. The magnitude of the impulse on \(B\) in the collision between \(B\) and \(C\) is \(3 m u\)
2. Find the value of \(k\).

1. Two particles, \(P\) and \(Q\), have masses \(m\) and \(e m\) respectively. The particles are moving on a smooth horizontal plane in the same direction along the same straight line when they collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(0 < e < 1\)

Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(e u\).

1. Show that the speed of \(Q\) immediately after the collision is \(u\).
2. Show that the direction of motion of \(P\) is unchanged by the collision. The magnitude of the impulse on \(Q\) in the collision is \(\frac { 2 } { 9 } m u\)
3. Find the possible values of \(e\).

1. Two particles, \(A\) and \(B\), have masses \(m\) and \(3 m\) respectively. The particles are moving in opposite directions along the same straight line on a smooth horizontal plane when they collide directly.

Immediately before they collide, \(A\) is moving with speed \(2 u\) and \(B\) is moving with speed \(u\). The direction of motion of each particle is reversed by the collision.
In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is \(\frac { 9 m u } { 2 }\)

1. Find the value of the coefficient of restitution between \(A\) and \(B\).
2. Hence, write down the total loss in kinetic energy due to the collision, giving a reason for your answer.

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 \(e\).

1. A particle \(P\), of mass 0.5 kg , is moving with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse I of magnitude 2.5 Ns.

As a result of the impulse, the direction of motion of \(P\) is deflected through an angle of \(45 ^ { \circ }\) Given that \(\mathbf { I } = ( \lambda \mathbf { i } + \mu \mathbf { j } )\) Ns, find all the possible pairs of values of \(\lambda\) and \(\mu\).

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 0.2 kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass 0.4 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 4 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) 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 \(\frac { 3 } { 7 }\)

1. Find the velocity of \(A\) immediately after the collision.
2. Find the magnitude of the impulse received by \(A\) in the collision.
3. Find, to the nearest degree, the size of the angle through which the direction of motion of \(A\) is deflected as a result of the collision.

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 \(P\) has mass 0.5 kg . It is moving in the \(x y\) plane with velocity \(8 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(\lambda ( - \mathbf { i } + \mathbf { j } )\) Ns, where \(\lambda\) is a positive constant.

The angle between the direction of motion of \(P\) immediately before receiving the impulse and the direction of motion of \(P\) immediately after receiving the impulse is \(\theta ^ { \circ }\) Immediately after receiving the impulse, \(P\) is moving with speed \(4 \sqrt { 10 } \mathrm {~ms} ^ { - 1 }\) Find (i) the value of \(\lambda\) (ii) the value of \(\theta\)

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

\begin{figure}[h] \end{figure} Figure 3 represents the plan view of part of a smooth horizontal floor, where \(A B\) is a fixed smooth vertical wall. The direction of \(\overrightarrow { A B }\) is in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\) A small ball of mass 0.25 kg is moving on the floor when it strikes the wall \(A B\).
Immediately before its impact with the wall \(A B\), the velocity of the ball is \(( 8 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Immediately after its impact with the wall \(A B\), the velocity of the ball is \(\mathbf { v m s } ^ { - 1 }\) The coefficient of restitution between the ball and the wall is \(\frac { 1 } { 3 }\) By modelling the ball as a particle,

1. show that \(\mathbf { v } = 4 \mathbf { i } + 6 \mathbf { j }\)
2. Find the magnitude of the impulse received by the ball in the impact.

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 \(k\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude 3 Ns .
The angle between the direction of motion of \(P\) immediately before receiving the impulse and the line of action of the impulse is \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. Find the speed of \(P\) immediately after receiving the impulse.

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.

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.

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

A particle \(A\) has mass 3 kg and a particle \(B\) has mass 2 kg .
The particles are moving on a smooth horizontal plane when they collide directly.
Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(( - 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Immediately after the collision the velocity of \(A\) is \(\left( - 2 \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\)

1. Find the total kinetic energy of the two particles before the collision.
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the impulse exerted on \(A\) by \(B\) in the collision.
3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after the collision.

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. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 2 \mathbf { i } - \mathbf { j } )\) Ns.

Show that the kinetic energy gained by \(P\) as a result of the impulse is 12 J .

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 \(u\).

3 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-2_473_298_1037_884} A particle \(P\) of mass 1.5 kg is attached to one end of a light inextensible string of length 2.4 m . The other end of the string is attached to a fixed point \(O\). The particle is initially at rest directly below \(O\). A horizontal impulse of magnitude 9.3 Ns is applied to \(P\). In the subsequent motion the string remains taut and makes an angle of \(\theta\) radians with the downwards vertical at \(O\), as shown in the diagram.

1. Find the speed of \(P\) when \(\theta = \frac { 1 } { 6 } \pi\).
2. Determine whether \(P\) will reach the same horizontal level as \(O\).

3 A body, \(Q\), of mass 2 kg moves in a straight line under the action of a single force which acts in the direction of motion of \(Q\). Initially the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\), the magnitude \(F N\) of the force is given by $$F = t ^ { 2 } + 3 \mathrm { e } ^ { t } , \quad 0 \leq t \leq 4 .$$

1. Calculate the impulse of the force over the time interval.
2. Hence find the speed of \(Q\) when \(t = 4\).

4 A particle of mass 0.5 kg is initially at rest. The particle then moves in a straight line under the action of a single force. This force acts in a constant direction and has magnitude \(\left( t ^ { 3 } + t \right) \mathrm { N }\), where \(t\) is the time, in seconds, for which the force has been acting.

1. Find the magnitude of the impulse exerted by the force on the particle between the times \(t = 0\) and \(t = 4\).
2. Hence find the speed of the particle when \(t = 4\).
3. Find the time taken for the particle to reach a speed of \(12 \mathrm {~ms} ^ { - 1 }\).

4 Two smooth spheres \(A\) and \(B\) of equal radius are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(m\) and \(4 m\) respectively.
The coefficient of restitution between the spheres is \(e\).
The spheres are projected directly towards each other, each with speed \(u\), and subsequently collide. 4

1. Show that the speed of \(B\) immediately after the impact with \(A\) is $$\frac { u ( 3 - 2 e ) } { 5 }$$ 4
2. Find the speed of \(A\) in terms of \(u\) and \(e\).
   4
3. Comment on the direction of motion of the spheres after the collision, justifying your answer.
   4
4. The magnitude of the impulse on \(B\) due to the collision is \(I\).
   Deduce that $$\frac { 8 m u } { 5 } \leq I \leq \frac { 16 m u } { 5 }$$