6.03a Linear momentum: p = mv

1 Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Write down the momentum of \(P\).
2. After the collision \(P\) continues to move in the same direction with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.

6. A railway truck \(P\) of mass 1500 kg is moving on a straight horizontal track. The truck \(P\) collides with a truck \(Q\) of 2500 kg at a point \(A\). Immediately before the collision, \(P\) and \(Q\) are moving in the same direction with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after the collision, the direction of motion of \(P\) is unchanged and its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By modelling the trucks as particles,

1. show that the speed of \(Q\) immediately after the collision is \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision at \(A\), the truck \(P\) is acted upon by a constant braking force of magnitude 500 N . The truck \(P\) comes to rest at the point \(B\).
2. Find the distance \(A B\). After the collision \(Q\) continues to move with constant speed \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance between \(P\) and \(Q\) at the instant when \(P\) comes to rest.

3. Two particles \(A\) and \(B\) have mass \(2 m\) and \(k m\) respectively. The particles are moving in opposite directions along the same straight smooth horizontal line so that the particles collide directly. Immediately before the collision \(A\) has speed \(2 u\) and \(B\) has speed \(u\). The direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of \(A\) is \(\frac { u } { 2 }\).

1. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(B\) on \(A\) in the collision.
2. Show that \(k < 5\)

3. Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(k m\), where \(2 < k < 3\), and the mass of \(B\) is \(m\). The particles are moving along the same straight line, but in opposite directions, and they collide directly. Immediately before they collide the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(4 u\). As a result of the collision the speed of \(A\) is halved and its direction of motion is reversed.

1. Find, in terms of \(k\) and \(u\), the speed of \(B\) immediately after the collision.
2. State whether the direction of motion of \(B\) changes as a result of the collision, explaining your answer. Given that \(k = \frac { 7 } { 3 }\),
3. find, in terms of \(m\) and \(u\), the magnitude of the impulse that \(A\) exerts on \(B\) in the collision.

1. Particles \(A , B\) and \(C\), of masses \(2 m , m\) and \(3 m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(2 u\) and collides directly with \(B\).

The coefficient of restitution between each pair of particles is \(e\).

1. 1. Show that the speed of \(B\) immediately after the collision with \(A\) is \(\frac { 4 } { 3 } u ( 1 + e )\)
   2. Find the speed of \(A\) immediately after the collision with \(B\). At the instant when \(A\) collides with \(B\), particle \(C\) is projected with speed \(u\) towards \(B\) so that \(B\) and \(C\) collide directly.
2. Show that there will be a second collision between \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-27_2644_1840_118_111}

6. Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface so that the particles collide directly.
The mass of \(P\) is \(k m\) and the mass of \(Q\) is \(m\).
Immediately before the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\). Immediately after the collision, \(P\) and \(Q\) are moving in the same direction, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(2 v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 5 }\) The magnitude of the impulse received by \(Q\) in the collision is \(5 m v\)

1. Find (i) \(y\) in terms of \(v\) (ii) \(x\) in terms of \(v\) (iii) the value of \(k\)
2. Find, in terms of \(m\) and \(v\), the total kinetic energy lost in the collision between \(P\) and \(Q\).

1. A particle \(P\) of mass \(3 m\) and a particle \(Q\) of mass \(5 m\) are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(k u\).
Immediately after the collision, the speed of \(P\) is \(2 v\) and the speed of \(Q\) is \(v\).
The direction of motion of each particle is reversed by the collision.
In the collision, \(P\) receives an impulse of magnitude \(15 m v\).

1. Show that \(u = 3 v\).
2. Find the value of \(k\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
3. Find the value of \(e\). The total kinetic energy lost in the collision is \(\lambda m v ^ { 2 }\)
4. Find the value of \(\lambda\).

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

A particle \(A\) has mass 2 kg and a particle \(B\) has mass 3 kg . The particles are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocity of \(A\) is \(5 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(B\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Immediately after the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

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 received by \(A\) in the collision. Given that, in the collision, the impulse of \(A\) on \(B\) is equal and opposite to the impulse of \(B\) on \(A\),
3. find the velocity of \(B\) immediately after the collision.

2. A particle \(P\) of mass 0.7 kg is moving in a straight line on a smooth horizontal surface. The particle \(P\) collides with a particle \(Q\) of mass 1.2 kg which is at rest on the surface. Immediately before the collision the speed of \(P\) is \(6 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision both particles are moving in the same direction. The coefficient of restitution between the particles is \(e\).

1. Show that \(e < \frac { 7 } { 12 }\) Given that \(e = \frac { 1 } { 4 }\)
2. find the magnitude of the impulse exerted on \(Q\) in the collision.
   

5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendi cular unit vectors in a horizontal plane.] A ball of mass 0.5 kg is moving with velocity \(( 10 \mathbf { i } + 24 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the impulse of the bat on the ball,
2. the size of the angle between the vector \(\mathbf { i }\) and the impulse exerted by the bat on the ball,
3. the kinetic energy lost by the ball in the impact.

8. A small ball A of mass 3 m is moving with speed u in a straight line on a smooth horizontal table. The ball collides directly with another small ball B of mass m moving with speed \(u\) towards \(A\) along the same straight line. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\). The balls have the same radius and can be modelled as particles.

1. Find
   1. the speed of A immediately after the collision,
   2. the speed of B immediately after the collision. A fter the collision \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\).
2. Find the speed of B immediately after hitting the wall.
   (2) The first collision between A and B occurred at a distance 4a from the wall. The balls collide again \(T\) seconds after the first collision.
3. Show that \(T = \frac { 112 a } { 15 u }\).

6. A smooth sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another smooth sphere \(B\) of mass \(3 m\), which is at rest on the table. The coefficient of restitution between \(A\) and \(B\) is \(e\). The spheres have the same radius and are modelled as particles.

1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 4 } ( 1 + e ) u\).
2. Find the speed of \(A\) immediately after the collision. Immediately after the collision the total kinetic energy of the spheres is \(\frac { 1 } { 6 } m u ^ { 2 }\).
3. Find the value of \(e\).
4. Hence show that \(A\) is at rest after the collision.

6. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal table. The particle \(P\) collides with a particle \(Q\) of mass \(2 m\) moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) after the collision is \(\frac { 1 } { 5 } u ( 9 e + 4 )\). As a result of the collision, the direction of motion of \(P\) is reversed.
2. Find the range of possible values of \(e\). Given that the magnitude of the impulse of \(P\) on \(Q\) is \(\frac { 32 } { 5 } m u\),
3. find the value of \(e\).
   (4)

1. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds,

$$\mathbf { F } = ( 6 t - 5 ) \mathbf { i } + \left( t ^ { 2 } - 2 t \right) \mathbf { j }$$ The velocity of \(P\) at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = \mathbf { i } - 4 \mathbf { j }\).

1. Find \(\mathbf { v }\) at time \(t\) seconds. When \(t = 3\), the particle \(P\) receives an impulse ( \(- 5 \mathbf { i } + 12 \mathbf { j }\) ) N s.
2. Find the speed of \(P\) immediately after it receives the impulse.

1. A particle of mass 0.25 kg is moving with velocity \(( 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives the impulse \(( 5 \mathbf { i } - 3 \mathbf { j } )\) N s.

Find the speed of the particle immediately after the impulse.

6 A particle \(P\) of mass 0.3 kg is projected upwards along a line of greatest slope from the foot of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The initial speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction is 0.15 . The particle \(P\) comes to instantaneous rest before it reaches the top of the plane.

1. Calculate the distance \(P\) moves up the plane.
2. Find the time taken by \(P\) to return from its highest position on the plane to the foot of the plane.
3. Calculate the change in the momentum of \(P\) between the instant that \(P\) leaves the foot of the plane and the instant that \(P\) returns to the foot of the plane.

5 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-3_291_182_799_945} Particles \(P\) and \(Q\), of masses 0.4 kg and \(m \mathrm {~kg}\) respectively, are joined by a light inextensible string which passes over a smooth pulley. The particles are released from rest at the same height above a horizontal surface; the string is taut and the portions of the string not in contact with the pulley are vertical (see diagram). \(Q\) begins to descend with acceleration \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and reaches the surface 0.3 s after being released. Subsequently, \(Q\) remains at rest and \(P\) never reaches the pulley.

1. Calculate the tension in the string while \(Q\) is in motion.
2. Calculate the momentum lost by \(Q\) when it reaches the surface.
3. Calculate the greatest height of \(P\) above the surface. \section*{[Questions 6 and 7 are printed overleaf.]}

7 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_122_255_1503_561} The diagram shows two particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, which move on a horizontal surface in the same direction along a straight line. A stationary particle \(R\) of mass 1.5 kg also lies on this line. \(P\) and \(Q\) collide and coalesce to form a combined particle \(C\). Immediately before this collision \(P\) has velocity \(4 \mathrm {~ms} ^ { - 1 }\) and \(Q\) has velocity \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the velocity of \(C\) immediately after this collision. At time \(t \mathrm {~s}\) after this collision the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(C\) is given by \(v = V _ { 0 } - 3 t ^ { 2 }\) for \(0 < t \leqslant 0.3\). \(C\) strikes \(R\) when \(t = 0.3\).
2. (a) State the value of \(V _ { 0 }\).
   (b) Calculate the distance \(C\) moves before it strikes \(R\).
   (c) Find the acceleration of \(C\) immediately before it strikes \(R\). Immediately after \(C\) strikes \(R\), the particles have equal speeds but move in opposite directions.
3. Find the speed of \(C\) immediately after it strikes \(R\).

1 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-2_135_917_274_575} Three particles \(P , Q\) and \(R\) have masses \(0.1 \mathrm {~kg} , 0.3 \mathrm {~kg}\) and 0.6 kg respectively. The particles travel along the same straight line on a smooth horizontal table and have velocities \(1.5 \mathrm {~ms} ^ { - 1 } , 1.1 \mathrm {~ms} ^ { - 1 }\) and \(0.8 \mathrm {~ms} ^ { - 1 }\) respectively (see diagram). \(P\) collides with \(Q\) and then \(Q\) collides with \(R\). In the second collision \(Q\) and \(R\) coalesce and subsequently move with a velocity of \(1 \mathrm {~ms} ^ { - 1 }\).

1. Find the speed of \(Q\) immediately before the second collision.
2. Calculate the change in momentum of \(P\) in the first collision.

2 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_138_1118_680_463} Three particles \(P , Q\) and \(R\) with masses \(0.4 \mathrm {~kg} , 0.3 \mathrm {~kg}\) and \(m \mathrm {~kg}\) are moving along the same straight line on a smooth horizontal surface. \(P\) and \(Q\) are moving towards each other with speeds \(u \mathrm {~ms} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(R\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the same direction as \(Q\) (see diagram).

1. Immediately after the collision between \(P\) and \(Q\) their directions of motion have been reversed, but their speeds are unchanged. Calculate \(u\). The next collision is between \(Q\) and \(R\). After the collision between \(Q\) and \(R\), particle \(Q\) is at rest and \(R\) has speed \(9 \mathrm {~ms} ^ { - 1 }\).
2. Calculate \(m\). \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_547_1506_1521_251} Two travellers \(A\) and \(B\) make the same journey on a long straight road. Each traveller walks for part of the journey and rides a bicycle for part of the journey. They start their journeys at the same instant, and they end their journeys simultaneously after travelling for \(T\) hours. \(A\) starts the journey cycling at a steady \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) for 1 hour. \(A\) then leaves the bicycle at the side of the road, and completes the journey walking at \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). \(B\) begins the journey walking at a steady \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). When \(B\) finds the bicycle where \(A\) left it, \(B\) cycles at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) to complete the journey (see diagram).