6.03b Conservation of momentum: 1D two particles

6 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. \(O\) is the centre of a vertical cross-section of the cylinder and \(D\) is the highest point on the cross-section. \(A\) and \(C\) are points on the circumference of the cross-section such that \(A O\) and \(C O\) are both inclined at an angle of \(30 ^ { \circ }\) below the horizontal diameter through \(O\). The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle \(P\), of mass \(m \mathrm {~kg}\), and a particle \(Q\), of mass \(5 m \mathrm {~kg}\), are simultaneously released from rest from \(A\) and \(C\), respectively, inside the cylinder. \(P\) and \(Q\) collide; the coefficient of restitution between them is 0.95 .

1. Show that, immediately after the collision, \(P\) moves with speed \(6.3 \mathrm {~ms} ^ { - 1 }\), and find the speed and direction of motion of \(Q\).
2. Find, in terms of \(m\), an expression for the normal reaction acting on \(P\) when it subsequently passes through \(D\).

2 \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-2_296_798_461_694} A sphere \(A\) of mass \(m\), moving on a horizontal surface, collides with another sphere \(B\) of mass \(2 m\), which is at rest on the surface. The spheres are smooth and uniform, and have equal radius. Immediately before the collision, \(A\) has velocity \(u\) at an angle \(\theta ^ { \circ }\) to the line of centres of the spheres (see diagram). Immediately after the collision, the spheres move in directions that are perpendicular to each other.

1. Find the coefficient of restitution between the spheres.
2. Given that the spheres have equal speeds after the collision, find \(\theta\).

4 A particle P of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point O . Particle P is projected so that it moves in complete vertical circles with centre O ; there is no air resistance. A and B are two points on the circle, situated on opposite sides of the vertical through O . The lines OA and OB make angles \(\alpha\) and \(\beta\) with the upward vertical as shown in Fig. 4. \begin{figure}[h] \end{figure} The speed of P at A is \(\sqrt { \frac { 17 a g } { 3 } }\). The speed of P at B is \(\sqrt { 5 a g }\) and \(\cos \beta = \frac { 2 } { 3 }\).

1. Show that \(\cos \alpha = \frac { 1 } { 3 }\). On one occasion, when P is at its lowest point and moving in a clockwise direction, it collides with a stationary particle Q . The two particles coalesce and the combined particle continues to move in the same vertical circle. When this combined particle reaches the point A , the string becomes slack.
2. Show that when the string becomes slack, the speed of the combined particle is \(\sqrt { \frac { a g } { 3 } }\). The mass of the particle Q is \(k m\).
3. Find the value of \(k\).
4. Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.

1. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 2 kg and sphere \(B\) has mass 3 kg . The spheres collide and at the instant of collision the line joining their centres is parallel to \(\mathbf { i }\). Before the collision \(A\) has velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision it has velocity \(( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Before the collision the velocity of \(B\) makes an angle \(\alpha\) with the line of centres, as shown in Fig. 1, where \(\tan \alpha = 2\). The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) before the collision.
(9)

2. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(3 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When the spheres collide, the line joining their centres is parallel to \(\mathbf { j }\), as shown in Figure 1.
The coefficient of restitution between the two spheres is \(e\).
The kinetic energy of sphere \(B\) immediately after the collision is \(85 \%\) of its kinetic energy immediately before the collision. Find

1. the velocity of each sphere immediately after the collision,
2. the value of \(e\).

3 A uniform rod, of mass 0.75 kg and length 1.6 m , rotates in a vertical plane about a fixed horizontal axis through one end. A frictional couple of constant moment opposes the motion. The rod is released from rest in a horizontal position and, when the rod is first vertical, its angular speed is \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the magnitude of the frictional couple. \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-2_584_527_1798_822} A disc is rotating about the same axis. The moment of inertia of the disc about the axis is \(0.56 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). When the rod is vertical, the disc has angular speed \(4.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in the opposite direction to that of the rod (see diagram). At this instant the rod hits a magnetic catch \(C\) on the disc and becomes attached to the disc.
2. Find the angular speed of the rod and disc immediately after they have become attached.

1 Two flywheels \(P\) and \(Q\) are rotating, in opposite directions, about the same fixed axis. The angular speed of \(P\) is \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the angular speed of \(Q\) is \(30 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The flywheels lock together, and after this they both rotate with angular speed \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in the direction in which \(P\) was originally rotating. The moment of inertia of \(P\) about the axis is \(0.64 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). Find the moment of inertia of \(Q\) about the axis.

1 Two flywheels \(F\) and \(G\) are rotating freely, about the same axis and in the same direction, with angular speeds \(21 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(36 \mathrm { rad } \mathrm { s } ^ { - 1 }\) respectively. The flywheels come into contact briefly, and immediately afterwards the angular speeds of \(F\) and \(G\) are \(28 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(34 \mathrm { rad } \mathrm { s } ^ { - 1 }\), respectively, in the same direction. Given that the moment of inertia of \(F\) about the axis is \(1.5 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), find the moment of inertia of \(G\) about the axis.

3 A circular disc is rotating in a horizontal plane with angular speed \(16 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis passing through its centre \(O\). The moment of inertia of the disc about the axis is \(0.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). A particle, initially at rest just above the surface of the disc, drops onto the disc and sticks to it at a point 0.4 m from \(O\). Afterwards, the angular speed of the disc with the particle attached is \(15 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle.
2. Find the loss of kinetic energy.

1 A uniform square lamina, of mass 4.5 kg and side 0.6 m , is rotating about a fixed vertical axis which is perpendicular to the lamina and passes through its centre. A stationary particle becomes attached to the lamina at one of its corners, and this causes the angular speed of the lamina to change instantaneously from \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle. The lamina then slows down with constant angular deceleration. It turns through 36 radians as its angular speed reduces from \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to zero.
2. Find the time taken for the lamina to come to rest.

2 A railway truck of mass \(m _ { 0 }\) travels along a horizontal track. There is no driving force and the resistances to motion are negligible. The truck is being filled with coal which falls vertically into it at a mass rate \(k\). The process starts as the truck passes a point O with speed \(u\). After time \(t\), the truck has velocity \(v\) and the displacement from O is \(x\).

1. Show that \(v = \frac { m _ { 0 } u } { m _ { 0 } + k t }\) and find \(x\) in terms of \(m _ { 0 } , u , k\) and \(t\).
2. Find the distance that the truck has travelled when its speed has been halved.

1 A rocket in deep space starts from rest and moves in a straight line. The initial mass of the rocket is \(m _ { 0 }\) and the propulsion system ejects matter at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. At time \(t\) the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket, \(\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k\).
2. Hence find an expression for \(v\) in terms of \(t\).
3. Find the speed of the rocket when its mass is \(\frac { 1 } { 3 } m _ { 0 }\).

1 A raindrop increases in mass as it falls vertically from rest through a stationary cloud. At time \(t \mathrm {~s}\) the velocity of the raindrop is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its mass is \(m \mathrm {~kg}\). The rate at which the mass increases is modelled as \(\frac { m g } { 2 ( v + 1 ) } \mathrm { kg } \mathrm { s } ^ { - 1 }\). Resistances to motion are neglected.

1. Write down the equation of motion of the raindrop. Hence show that $$\left( 1 - \frac { 1 } { v + 2 } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 2 } g .$$
2. Solve this differential equation to find an expression for \(t\) in terms of \(v\). Calculate the time it takes for the velocity of the raindrop to reach \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Describe, with reasons, what happens to the acceleration of the raindrop for large values of \(t\).

1 At time \(t\) a rocket has mass \(m\) and is moving vertically upwards with velocity \(v\). The propulsion system ejects matter at a constant speed \(u\) relative to the rocket. The only additional force acting on the rocket is its weight.

1. Derive the differential equation \(m \frac { \mathrm {~d} v } { \mathrm {~d} t } + u \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g\). The rocket has initial mass \(m _ { 0 }\) of which \(75 \%\) is fuel. It is launched from rest. Matter is ejected at a constant mass rate \(k\).
2. Assuming that the acceleration due to gravity is constant, find the speed of the rocket at the instant when all the fuel is burnt.

1 A raindrop of mass \(m\) falls vertically from rest under gravity. Initially the mass of the raindrop is \(m _ { 0 }\). As it falls it loses mass by evaporation at a rate \(\lambda m\), where \(\lambda\) is a constant. Its motion is modelled by assuming that the evaporation produces no resultant force on the raindrop. The velocity of the raindrop is \(v\) at time \(t\). The forces on the raindrop are its weight and a resistance force of magnitude \(k m v\), where \(k\) is a constant.

1. Find \(m\) in terms of \(m _ { 0 } , \lambda\) and \(t\).
2. Write down the equation of motion of the raindrop. Solve this differential equation and hence show that \(v = \frac { g } { \lambda - k } \left( \mathrm { e } ^ { ( \lambda - k ) t } - 1 \right)\).
3. Find the velocity of the raindrop when it has lost half of its initial mass.

1 A rocket in deep space has initial mass \(m _ { 0 }\) and is moving in a straight line at speed \(v _ { 0 }\). It fires its engine in the direction opposite to the motion in order to increase its speed. The propulsion system ejects matter at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. At time \(t\) after the engines are fired, the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket, \(\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k\).
2. Hence find an expression for \(v\) at time \(t\).

1 An empty railway truck of mass \(m _ { 0 }\) is moving along a straight horizontal track at speed \(v _ { 0 }\). The point P is at the front of the truck. The horizontal forces on the truck are negligible. As P passes a fixed point O , sand starts to fall vertically into the truck at a constant mass rate \(k\). At time \(t\) after P passes O the speed of the truck is \(v\) and \(\mathrm { OP } = x\).

1. Find an expression for \(v\) in terms of \(m _ { 0 } , v _ { 0 } , k\) and \(t\), and show that \(x = \frac { m _ { 0 } v _ { 0 } } { k } \ln \left( 1 + \frac { k t } { m _ { 0 } } \right)\).
2. Find the speed of the truck and the distance OP when the mass of sand in the truck is \(2 m _ { 0 }\).

2 On a building site, a pulley system is used for moving small amounts of material up to roof level. A light pulley, which can rotate freely, is attached with its axis horizontal to the top of some scaffolding. A light inextensible rope hangs over the pulley with a counterweight of mass \(m _ { 1 } \mathrm {~kg}\) attached to one end. Attached to the other end of the rope is a bag of negligible mass into which \(m _ { 2 } \mathrm {~kg}\) of roof tiles are placed, where \(m _ { 2 } < m _ { 1 }\). This situation is shown in Fig. 2. \begin{figure}[h] \end{figure} Initially the system is held at rest with the rope taut, the counterweight at the top of the scaffolding and the bag of tiles on the ground. When the counterweight is released, the bag ascends towards the top of the scaffolding. At time \(t \mathrm {~s}\) the velocity of the counterweight is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards. The counterweight is made from a bag of negligible mass filled with sand. At the moment the counterweight is released, this bag is accidentally ripped and after this time the sand drops out at a constant rate of \(\lambda \mathrm { kg } \mathrm { s } ^ { - 1 }\).

1. Find the equation of motion for the counterweight while it still contains sand, and hence show that $$v = g t + \frac { 2 g m _ { 2 } } { \lambda } \ln \left( 1 - \frac { \lambda t } { m _ { 1 } + m _ { 2 } } \right) .$$
2. Given that the sand would run out after 10 seconds and that \(m _ { 2 } = \frac { 4 } { 5 } m _ { 1 }\), find the maximum velocity attained by the counterweight towards the ground. You may assume that the scaffolding is sufficiently high that the counterweight does not hit the ground before this velocity is reached.

4. A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane. The rod hangs in equilibrium with \(B\) below \(A\). The rod is rotated through a small angle and released from rest at time \(t = 0\).

1. Show that the motion of the rod is approximately simple harmonic.
2. Using this approximation, find the time \(t\) when the rod is first vertical after being released.
   (Total 6 marks)

6. The vertices of a tetrahedron \(P Q R S\) have position vectors \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and \(\mathbf { s }\) respectively, where $$\mathbf { p } = - 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad \mathbf { q } = 4 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \mathbf { s } = 4 \mathbf { i } + \mathbf { k }$$ Forces of magnitude 20 N and \(2 \sqrt { } 13 \mathrm {~N}\) act along \(S Q\) and \(S R\) respectively. A third force acts at \(P\).
Given that the system of three forces reduces to a couple \(\mathbf { G }\), find

1. the third force,
2. the magnitude of \(\mathbf { G }\).
   (6)
   (Total 12 marks)

7. A uniform plane circular disc, of mass \(m\) and radius \(a\), hangs in equilibrium from a point \(B\) on its circumference. The disc is free to rotate about a fixed smooth horizontal axis which is in the plane of the disc and tangential to the disc at \(B\). A particle \(P\), of mass \(m\), is moving horizontally with speed \(u\) in a direction which is perpendicular to the plane of the disc. At time \(t = 0 , P\) strikes the disc at its centre and adheres to the disc.

1. Show that the angular speed of the disc immediately after it has been struck by \(P\) is \(\frac { 4 u } { 9 a }\).
   (6) It is given that \(u ^ { 2 } = \frac { 1 } { 10 } a g\), and that air resistance is negligible.
2. Find the angle through which the disc turns before it first comes to instantaneous rest. The disc first returns to its initial position at time \(t = T\).
3. 1. Write down an equation of motion for the disc.
   2. Hence find \(T\) in terms of \(a , g\) and \(m\), using a suitable approximation which should be justified.

5. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The rod is hanging in equilibrium with \(B\) below \(A\) when it is hit by a particle of mass \(m\) moving horizontally with speed \(v\) in a vertical plane perpendicular to the axis. The particle strikes the rod at \(B\) and immediately adheres to it.

1. Show that the angular speed of the rod immediately after the impact is \(\frac { 3 v } { 8 a }\). Given that the rod rotates through \(120 ^ { \circ }\) before first coming to instantaneous rest,
2. find \(v\) in terms of \(a\) and \(g\).
3. find, in terms of \(m\) and \(g\), the magnitude of the vertical component of the force acting on the \(\operatorname { rod }\) at \(A\) immediately after the impact.
   (5)

5. A rocket is launched vertically upwards under gravity from rest at time \(t = 0\). The rocket propels itself upward by ejecting burnt fuel vertically downwards at a constant speed \(u\) relative to the rocket. The initial mass of the rocket, including fuel, is \(M\). At time \(t\), before all the fuel has been used up, the mass of the rocket, including fuel, is \(M ( 1 - k t )\) and the speed of the rocket is \(v\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { k u } { 1 - k t } - g\).
2. Hence find the speed of the rocket when \(t = \frac { 1 } { 3 k }\).

6. A rocket-driven car moves along a straight horizontal road. The car has total initial mass \(M\). It propels itself forwards by ejecting mass backwards at a constant rate \(\lambda\) per unit time at a constant speed \(U\) relative to the car. The car starts from rest at time \(t = 0\). At time \(t\) the speed of the car is \(v\). The total resistance to motion is modelled as having magnitude \(k v\), where \(k\) is a constant. Given that \(t < \frac { M } { \lambda }\), show that

1. \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }\),
2. \(v = \frac { \lambda U } { k } \left\{ 1 - \left( 1 - \frac { \lambda t } { M } \right) ^ { \frac { k } { \lambda } } \right\}\).
   (6)
   (Total 13 marks)

7. A motor boat of mass \(M\) is moving in a straight line, with its engine switched off, across a stretch of still water. The boat is moving with speed \(U\) when, at time \(t = 0\), it develops a leak. The water comes in at a constant rate so that at time \(t\), the mass of water in the boat is \(\lambda t\). At time \(t\) the speed of the boat is \(v\) and it experiences a total resistance to motion of magnitude \(2 \lambda v\).

1. Show that \(( M + \lambda t ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 \lambda v = 0\).
   (6)
2. Show that the time taken for the speed of the boat to reduce to \(\frac { 1 } { 2 } U\) is \(\frac { M } { \lambda } \left( 2 ^ { \frac { 1 } { 3 } } - 1 \right)\).
   (6) The boat sinks when the mass of water inside the boat is \(M\).
3. Show that the boat does not sink before the speed of the boat is \(\frac { 1 } { 2 } U\).