6.03b Conservation of momentum: 1D two particles

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5m\) and \(2m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2u\). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac{1}{7}u(1 + 15e)\) and find an expression for the speed of \(A\). [4]

In the collision, the speed of \(A\) is halved and its direction of motion is reversed.

1. Find the value of \(e\). [2]
2. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\). [3]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(2m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac{2}{3}\).

1. Find, in terms of \(u\), the speeds of \(A\) and \(B\) after this collision. [4]
2. Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac{1}{2}\). Find, in terms of \(d\) and \(u\), the time that elapses between the first and second collisions between \(A\) and \(B\). [5]

Three uniform small spheres \(A\), \(B\) and \(C\) have equal radii and masses \(5m\), \(5m\) and \(3m\) respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\).

1. Show that the speed of \(A\) after its collision with \(B\) is \(\frac{1}{2}u(1 - e)\) and find the speed of \(B\). [3]

Sphere \(B\) now collides with sphere \(C\). Subsequently there are no further collisions between any of the spheres.

1. Find the set of possible values of \(e\). [6]

A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]

Small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(10 \text{ ms}^{-1}\). After the spheres collide \(A\) continues to move in the same direction but with half the speed of \(B\).

1. Find the speed of \(B\) after the collision. [2]

A third small smooth sphere \(C\), of mass 1 kg and with the same radius as \(A\) and \(B\), is at rest on the plane. \(B\) now collides directly with \(C\). After this collision \(B\) continues to move in the same direction but with one third the speed of \(C\).

1. Show that there is another collision between \(A\) and \(B\). [3]
2. \(A\) and \(B\) coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions. [5]

Three particles \(P\), \(Q\) and \(R\), of masses 0.1 kg, 0.2 kg and 0.5 kg respectively, are at rest in a straight line on a smooth horizontal plane. Particle \(P\) is projected towards \(Q\) at a speed of \(5 \text{ m s}^{-1}\). After \(P\) and \(Q\) collide, \(P\) rebounds with speed \(1 \text{ m s}^{-1}\).

1. Find the speed of \(Q\) immediately after the collision with \(P\). [3]

\(Q\) now collides with \(R\). Immediately after the collision with \(Q\), \(R\) begins to move with speed \(V \text{ m s}^{-1}\).

1. Given that there is no subsequent collision between \(P\) and \(Q\), find the greatest possible value of \(V\). [3]

Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at 30° to the horizontal, and \(A\) is higher up the plane than \(B\). When the particles collide, the speeds of \(A\) and \(B\) are 3 m s\(^{-1}\) and 2 m s\(^{-1}\) respectively. In the collision between the particles, the speed of \(A\) is reduced to 2.5 m s\(^{-1}\).

1. Find the speed of \(B\) immediately after the collision. [2]

After the collision, when \(B\) has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. \(B\) hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by 90%. The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.

1. Show that the speed of \(B\) immediately after it hits the barrier is 0.5 m s\(^{-1}\). Hence find the speed of the combined particle immediately after the second collision between \(A\) and \(B\). [7]

Small smooth spheres \(A\) and \(B\), of equal radii and of masses \(5\text{kg}\) and \(3\text{kg}\) respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(8.5\text{ms}^{-1}\). The spheres collide and after the collision \(A\) continues to move in the same direction but with a quarter of the speed of \(B\).

1. Find the speed of \(B\) after the collision. [3]
2. Find the loss of kinetic energy of the system due to the collision. [2]

Two particles \(P\) and \(Q\), of masses \(m\) kg and \(0.3\) kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(5\) m s\(^{-1}\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(P\) moves with a speed of \(2\) m s\(^{-1}\) in the same direction as it was originally moving.

1. Find, in terms of \(m\), the speed of \(Q\) after the collision. [2]

After this collision, \(Q\) moves directly towards a third particle \(R\), of mass \(0.6\) kg, which is at rest on the plane. \(Q\) is brought to rest in the collision with \(R\), and \(R\) begins to move with a speed of \(1.5\) m s\(^{-1}\).

1. Find the value of \(m\). [2]

Three particles \(A\), \(B\) and \(C\) of masses 5 kg, 1 kg and 2 kg respectively lie at rest in that order on a straight smooth horizontal track \(XYZ\). Initially \(A\) is at \(X\), \(B\) is at \(Y\) and \(C\) is at \(Z\). Particle \(A\) is projected towards \(B\) with a speed of \(6\text{ ms}^{-1}\) and at the same instant \(C\) is projected towards \(B\) with a speed of \(v\text{ ms}^{-1}\). In the subsequent motion, \(A\) collides and coalesces with \(B\) to form particle \(D\). Particle \(D\) then collides and coalesces with \(C\) to form particle \(E\) and \(E\) moves towards \(Z\).

1. Show that after the second collision the speed of \(E\) is \(\frac{15-v}{4}\text{ ms}^{-1}\). [3]
2. The total loss of kinetic energy of the system due to the two collisions is 63 J. Use the result from (a) to show that \(v = 3\). [3]
3. It is given that the distance \(XY\) is 36 m and the distance \(YZ\) is 98 m.
   1. Find the time between the two collisions. [4]
   2. Find the time between the instant that \(A\) is projected from \(X\) and the instant that \(E\) reaches \(Z\). [1]

\includegraphics{figure_7} The diagram shows a smooth track which lies in a vertical plane. The section \(AB\) is a quarter circle of radius 1.8 m with centre \(O\). The section \(BC\) is a horizontal straight line of length 7.0 m and \(OB\) is perpendicular to \(BC\). The section \(CFE\) is a straight line inclined at an angle of \(\theta°\) above the horizontal. A particle \(P\) of mass 0.5 kg is released from rest at \(A\). Particle \(P\) collides with a particle \(Q\) of mass 0.1 kg which is at rest at \(B\). Immediately after the collision, the speed of \(P\) is \(4\,\text{m}\,\text{s}^{-1}\) in the direction \(BC\). You should assume that \(P\) is moving horizontally when it collides with \(Q\).

1. Show that the speed of \(Q\) immediately after the collision is \(10\,\text{m}\,\text{s}^{-1}\). [4] When \(Q\) reaches \(C\), it collides with a particle \(R\) of mass 0.4 kg which is at rest at \(C\). The two particles coalesce. The combined particle comes instantaneously to rest at \(F\). You should assume that there is no instantaneous change in speed as the combined particle leaves \(C\), nor when it passes through \(C\) again as it returns down the slope.
2. Given that the distance \(CF\) is 0.4 m, find the value of \(\theta\). [4]
3. Find the distance from \(B\) at which \(P\) collides with the combined particle. [5]

\includegraphics{figure_7} The diagram shows two particles \(P\) and \(Q\) which lie on a line of greatest slope of a plane \(ABC\). Particles \(P\) and \(Q\) are each of mass \(m\) kg. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.6\). The length of \(AB\) is 0.75 m and the length of \(BC\) is 3.25 m. The section \(AB\) of the plane is smooth and the section \(BC\) is rough. The coefficient of friction between each particle and the section \(BC\) is 0.25. Particle \(P\) is released from rest at \(A\). At the same instant, particle \(Q\) is released from rest at \(B\).

1. Verify that particle \(P\) reaches \(B\) 0.5 s after it is released, with speed \(3\) m s\(^{-1}\). [3]
2. Find the time that it takes from the instant the two particles are released until they collide. [4]

The two particles coalesce when they collide. The coefficient of friction between the combined particle and the plane is still 0.25.

1. Find the time that it takes from the instant the particles collide until the combined particle reaches \(C\). [5]

A particle \(B\) of mass 5 kg is at rest on a smooth horizontal table. A particle \(A\) of mass 2.5 kg moves on the table with a speed of \(6 \text{ m s}^{-1}\) and collides directly with \(B\). In the collision the two particles coalesce.

1. Find the speed of the combined particle after the collision. [2]
2. Find the loss of kinetic energy of the system due to the collision. [3]

Small smooth spheres \(A\) and \(B\), of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(A\) is moving towards \(B\) with speed 5 m s\(^{-1}\) and \(B\) is moving towards \(A\) with speed 3 m s\(^{-1}\). After the spheres collide, both \(A\) and \(B\) move in the same direction and the difference in the speeds of the spheres is 2 m s\(^{-1}\). Find the loss of kinetic energy of the system due to the collision. [5]

Three particles \(A\), \(B\) and \(C\) of masses 0.3 kg, 0.4 kg and \(m\) kg respectively lie at rest in a straight line on a smooth horizontal plane. The distance between \(B\) and \(C\) is 2.1 m. \(A\) is projected directly towards \(B\) with speed \(2 \text{ m s}^{-1}\). After \(A\) collides with \(B\) the speed of \(A\) is reduced to \(0.6 \text{ m s}^{-1}\), still moving in the same direction.

1. Show that the speed of \(B\) after the collision is \(1.05 \text{ m s}^{-1}\). [2]

After the collision between \(A\) and \(B\), \(B\) moves directly towards \(C\). Particle \(B\) now collides with \(C\). After this collision, the two particles coalesce and have a combined speed of \(0.5 \text{ m s}^{-1}\).

1. Find \(m\). [2]

1. Find the time that it takes, from the instant when \(B\) and \(C\) collide, until \(A\) collides with the combined particle. [5]

A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed 25 m s\(^{-1}\).

1. Find the speed of \(A\) when it reaches a height of 20 m above the ground. [2]

When \(A\) reaches a height of 20 m, it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed 32.5 m s\(^{-1}\). In the collision between the two particles, \(B\) is brought to instantaneous rest.

1. Show that the velocity of \(A\) immediately after the collision is 4.5 m s\(^{-1}\) downwards. [2]
2. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground. [4]

A particle, \(A\), is projected vertically upwards from a point \(O\) with a speed of \(80 \text{ ms}^{-1}\). One second later a second particle, \(B\), with the same mass as \(A\), is projected vertically upwards from \(O\) with a speed of \(100 \text{ ms}^{-1}\). At time \(T\) s after the first particle is projected, the two particles collide and coalesce to form a particle \(C\).

1. Show that \(T = 3.5\). [4]
2. Find the height above \(O\) at which the particles collide. [1]
3. Find the time from \(A\) being projected until \(C\) returns to \(O\). [5]

Two particles, \(A\) and \(B\), of masses 3 kg and 6 kg respectively, lie on a smooth horizontal plane. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with speed 8 ms\(^{-1}\). After \(A\) and \(B\) collide, \(A\) moves with speed 2 ms\(^{-1}\). Find the greater of the two possible total losses of kinetic energy due to the collision. [6]

A particle \(P\) is attached to one end of a light elastic string of natural length \(1.2 \text{ m}\) and modulus of elasticity \(12 \text{ N}\). The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. \(P\) rests in equilibrium on the plane, \(1.6 \text{ m}\) from \(O\).

1. Calculate the mass of \(P\). [2]

A particle \(Q\), with mass equal to the mass of \(P\), is projected up the plane along a line of greatest slope. When \(Q\) strikes \(P\) the two particles coalesce. The combined particle remains attached to the string and moves up the plane, coming to instantaneous rest after moving \(0.2 \text{ m}\).

1. Show that the initial kinetic energy of the combined particle is \(1 \text{ J}\). [4]

The combined particle subsequently moves down the plane.

1. Calculate the greatest speed of the combined particle in the subsequent motion. [5]

A particle \(P\) is attached to one end of a light elastic string of natural length \(1.2\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. \(P\) rests in equilibrium on the plane, \(1.6\) m from \(O\).

1. Calculate the mass of \(P\). [2]

A particle \(Q\), with mass equal to the mass of \(P\), is projected up the plane along a line of greatest slope. When \(Q\) strikes \(P\) the two particles coalesce. The combined particle remains attached to the string and moves up the plane, coming to instantaneous rest after moving \(0.2\) m.

1. Show that the initial kinetic energy of the combined particle is \(1\) J. [4]

The combined particle subsequently moves down the plane.

1. Calculate the greatest speed of the combined particle in the subsequent motion. [5]

Two particles \(A\) and \(B\) have masses \(3m\) and \(2m\) respectively. Initially \(A\) is at rest and \(B\) is moving with speed \(u\) in a straight line towards \(A\). The coefficient of restitution between the particles is \(e\).

1. Find the speeds of the particles immediately after the collision.
2. Find the condition on \(e\) for \(A\) to be moving faster than \(B\) after the collision.

[8]

\includegraphics{figure_7} One end of a light spring of natural length \(a\) and modulus of elasticity \(4mg\) is attached to a fixed point \(O\). The other end of the spring is attached to a particle \(A\) of mass \(km\), where \(k\) is a constant. Initially the spring lies at rest on a smooth horizontal surface and has length \(a\). A second particle \(B\), of mass \(m\), is moving towards \(A\) with speed \(\sqrt{\frac{4}{3}ga}\) along the line of the spring from the opposite direction to \(O\) (see diagram). The particles \(A\) and \(B\) collide and coalesce. At a point \(C\) in the subsequent motion, the length of the spring is \(\frac{5}{4}a\) and the speed of the combined particle is half of its initial speed.

1. Find the value of \(k\). [6]

A light spring \(AB\) has natural length \(a\) and modulus of elasticity \(5mg\). The end \(A\) of the spring is attached to a fixed point on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the end \(B\) of the spring. The spring and particle \(P\) are at rest on the surface. Another particle \(Q\) of mass \(km\) is moving with speed \(\sqrt{4ga}\) along the horizontal surface towards \(P\) in the direction \(BA\). The particles \(P\) and \(Q\) collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is \(\frac{2}{3}a\). Find the value of \(k\). [6]

A railway truck \(A\) of mass \(m\) and a second railway truck \(B\) of mass \(4m\) are moving in opposite directions on a smooth straight horizontal track when they collide directly. Immediately before the collision the speed of truck \(A\) is \(3u\) and the speed of truck \(B\) is \(2u\). In the collision the trucks join together. Modelling the trucks as particles, find

1. the speed of \(A\) immediately after the collision, [3]
2. the direction of motion of \(A\) immediately after the collision, [1]
3. the magnitude of the impulse exerted by \(A\) on \(B\) in the collision. [3]

Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) is moving due east and particle \(Q\) is moving due west. Particle \(P\) has mass \(2m\) and particle \(Q\) has mass \(3m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(4u\) and the speed of \(Q\) is \(u\). The magnitude of the impulse in the collision is \(\frac{33}{5}mu\).

1. Find the speed and direction of motion of \(P\) immediately after the collision. [4]
2. Find the speed and direction of motion of \(Q\) immediately after the collision. [4]