6.03f Impulse-momentum: relation

Two trucks \(A\) and \(B\), moving in opposite directions on the same horizontal railway track, collide. The mass of \(A\) is 600 kg. The mass of \(B\) is \(m\) kg. Immediately before the collision, the speed of \(A\) is 4 m s\(^{-1}\) and the speed of \(B\) is 2 m s\(^{-1}\). Immediately after the collision, the trucks are joined together and move with the same speed 0.5 m s\(^{-1}\). The direction of motion of \(A\) is unchanged by the collision. Find

1. the value of \(m\), [4]
2. the magnitude of the impulse exerted on \(A\) in the collision. [3]

A particle \(P\) of mass 1.5 kg is moving along a straight horizontal line with speed 3 m s\(^{-1}\). Another particle \(Q\) of mass 2.5 kg is moving, in the opposite direction, along the same straight line with speed 4 m s\(^{-1}\). The particles collide. Immediately after the collision the direction of motion of \(P\) is reversed and its speed is 2.5 m s\(^{-1}\).

1. Calculate the speed of \(Q\) immediately after the impact. [3]
2. State whether or not the direction of motion of \(Q\) is changed by the collision. [1]
3. Calculate the magnitude of the impulse exerted by \(Q\) on \(P\), giving the units of your answer. [3]

A stone \(S\) is sliding on ice. The stone is moving along a straight horizontal line \(ABC\), where \(AB = 24\) m and \(AC = 30\) m. The stone is subject to a constant resistance to motion of magnitude 0.3 N. At \(A\) the speed of \(S\) is 20 m s\(^{-1}\), and at \(B\) the speed of \(S\) is 16 m s\(^{-1}\). Calculate

1. the deceleration of \(S\), [2]
2. the speed of \(S\) at \(C\). [3]
3. Show that the mass of \(S\) is 0.1 kg. [2]

At \(C\), the stone \(S\) hits a vertical wall, rebounds from the wall and then slides back along the line \(CA\). The magnitude of the impulse of the wall on \(S\) is 2.4 Ns and the stone continues to move against a constant resistance of 0.3 N.

1. Calculate the time between the instant that \(S\) rebounds from the wall and the instant that \(S\) comes to rest. [6]

A particle \(P\) of mass 0.3 kg is moving with speed \(u\) m s\(^{-1}\) in a straight line on a smooth horizontal table. The particle \(P\) collides directly with a particle \(Q\) of mass 0.6 kg, which is at rest on the table. Immediately after the particles collide, \(P\) has speed 2 m s\(^{-1}\) and \(Q\) has speed 5 m s\(^{-1}\). The direction of motion of \(P\) is reversed by the collision. Find

1. the value of \(u\), [4]
2. the magnitude of the impulse exerted by \(P\) on \(Q\). [2]

Immediately after the collision, a constant force of magnitude \(R\) newtons is applied to \(Q\) in the direction directly opposite to the direction of motion of \(Q\). As a result \(Q\) is brought to rest in 1.5 s.

1. Find the value of \(R\). [4]

A particle \(A\) of mass 2 kg is moving along a straight horizontal line with speed 12 m s\(^{-1}\). Another particle \(B\) of mass \(m\) kg is moving along the same straight line, in the opposite direction to \(A\), with speed 8 m s\(^{-1}\). The particles collide. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, \(A\) is moving with speed 3 m s\(^{-1}\) and \(B\) is moving with speed 4 m s\(^{-1}\). Find

1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision, [2]
2. the value of \(m\). [4]

Two particles \(P\) and \(Q\) have masses \(4m\) and \(m\) respectively. The particles are moving towards each other on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2u\) and \(5u\) respectively. Immediately after the collision, the speed of \(P\) is \(\frac{1}{2}u\) and its direction of motion is reversed.

1. Find the speed and direction of motion of \(Q\) after the collision. [4]
2. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision. [3]

The masses of two particles \(A\) and \(B\) are \(0.5 \text{ kg}\) and \(m \text{ kg}\) respectively. The particles are moving on a smooth horizontal table in opposite directions and collide directly. Immediately before the collision the speed of \(A\) is \(5 \text{ m s}^{-1}\) and the speed of \(B\) is \(3 \text{ m s}^{-1}\). In the collision, the magnitude of the impulse exerted by \(B\) on \(A\) is \(3.6 \text{ Ns}\). As a result of the collision the direction of motion of \(A\) is reversed.

1. Find the speed of \(A\) immediately after the collision. [3]

The speed of \(B\) immediately after the collision is \(1 \text{ m s}^{-1}\).

1. Find the two possible values of \(m\). [4]

A particle \(P\) is moving with constant acceleration along a straight horizontal line \(ABC\), where \(AC = 24\) m. Initially \(P\) is at \(A\) and is moving with speed \(5\) m s\(^{-1}\) in the direction \(AB\). After \(1.5\) s, the direction of motion of \(P\) is unchanged and \(P\) is at \(B\) with speed \(9.5\) m s\(^{-1}\).

1. Show that the speed of \(P\) at \(C\) is \(13\) m s\(^{-1}\). [4]

The mass of \(P\) is \(2\) kg. When \(P\) reaches \(C\), an impulse of magnitude \(30\) Ns is applied to \(P\) in the direction \(CB\).

1. Find the velocity of \(P\) immediately after the impulse has been applied, stating clearly the direction of motion of \(P\) at this instant. [3]

Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \text{ m s}^{-1}\) and the speed of \(B\) is \(2 \text{ m s}^{-1}\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find

1. the speed of \(A\) immediately after the collision, [5]
2. the magnitude of the impulse exerted on \(B\) in the collision. [3]

Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(2m\) 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 \(2u\) and the speed of \(B\) is \(3u\). The magnitude of the impulse received by each particle in the collision is \(\frac{7mu}{2}\). Find

1. the speed of \(A\) immediately after the collision, [3]
2. the speed of \(B\) immediately after the collision. [3]

Particle \(P\) has mass 3 kg and particle \(Q\) has mass 2 kg. The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, \(P\) has speed 3 m s\(^{-1}\) and \(Q\) has speed 2 m s\(^{-1}\). Immediately after the collision, both particles move in the same direction and the difference in their speeds is 1 m s\(^{-1}\).

1. Find the speed of each particle after the collision. [5]
2. Find the magnitude of the impulse exerted on \(P\) by \(Q\). [3]

Two particles \(A\) and \(B\), of mass 2 kg and 3 kg respectively, are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately before the collision the speed of \(A\) is 5 m s\(^{-1}\) and the speed of \(B\) is 6 m s\(^{-1}\). The magnitude of the impulse exerted on \(B\) by \(A\) is 14 N s. Find

1. the speed of \(A\) immediately after the collision, [3]
2. the speed of \(B\) immediately after the collision. [3]

Particle \(P\) has mass 3 kg and particle \(Q\) has mass \(m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of \(P\) is \(4 \text{ m s}^{-1}\) and the speed of \(Q\) is \(3 \text{ m s}^{-1}\). In the collision the direction of motion of \(P\) is unchanged and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(1 \text{ m s}^{-1}\) and the speed of \(Q\) is \(1.5 \text{ m s}^{-1}\).

1. Find the magnitude of the impulse exerted on \(P\) in the collision. [3]
2. Find the value of \(m\). [3]

A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of \(1.6\) m above the top of the post.

1. Show that the speed of the pile-driver just before it hits the post is \(5.6\) m s\(^{-1}\). [2]

The post has mass \(6\) kg and the pile-driver has mass \(78\) kg. When the pile-driver hits the top of the post, it is assumed that the there is no rebound and that both then move together with the same speed.

1. Find the speed of the pile-driver and the post immediately after the pile-driver has hit the post. [3]

The post is brought to rest by the action of a resistive force from the ground acting for \(0.06\) s. By modelling this force as constant throughout this time,

1. find the magnitude of the resistive force, [4]
2. find, to 2 significant figures, the distance travelled by the post and the pile-driver before they come to rest. [4]

A railway truck \(S\) of mass 2000 kg is travelling due east along a straight horizontal track with constant speed 12 m s\(^{-1}\). The truck \(S\) collides with a truck \(T\) which is travelling due west along the same track as \(S\) with constant speed 6 m s\(^{-1}\). The magnitude of the impulse of \(T\) on \(S\) is 28800 Ns.

1. Calculate the speed of \(S\) immediately after the collision. [3]
2. State the direction of motion of \(S\) immediately after the collision. [1]

Given that, immediately after the collision, the speed of \(T\) is 3.6 m s\(^{-1}\), and that \(T\) and \(S\) are moving in opposite directions,

1. calculate the mass of \(T\). [4]

A tent peg is driven into soft ground by a blow from a hammer. The tent peg has mass 0.2 kg and the hammer has mass 3 kg. The hammer strikes the peg vertically. Immediately before the impact, the speed of the hammer is \(16 \text{ m s}^{-1}\). It is assumed that, immediately after the impact, the hammer and the peg move together vertically downwards.

1. Find the common speed of the peg and the hammer immediately after the impact. [3]

Until the peg and hammer come to rest, the resistance exerted by the ground is assumed to be constant and of magnitude \(R\) newtons. The hammer and peg are brought to rest 0.05 s after the impact.

1. Find, to 3 significant figures, the value of \(R\). [5]

A truck of mass 3 tonnes moves on straight horizontal rails. It collides with truck \(B\) of mass 1 tonne, which is moving on the same rails. Immediately before the collision, the speed of \(A\) is \(3 \text{ m s}^{-1}\), the speed of \(B\) is \(4 \text{ m s}^{-1}\), and the trucks are moving towards each other. In the collision, the trucks couple to form a single body \(C\), which continues to move on the rails.

1. Find the speed and direction of \(C\) after the collision. [4]
2. Find, in Ns, the magnitude of the impulse exerted by \(B\) on \(A\) in the collision. [3]
3. State a modelling assumption which you have made about the trucks in your solution [1]

Immediately after the collision, a constant braking force of magnitude 250 N is applied to \(C\). It comes to rest in a distance \(d\) metres.

1. Find the value of \(d\). [4]

A particle \(P\) of mass 2 kg is moving with velocity \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) when it receives an impulse. Immediately after the impulse is applied, \(P\) has velocity \((2\mathbf{i} - 3\mathbf{j})\) m s\(^{-1}\).

1. Find the magnitude of the impulse. [5]
2. Find the angle between the direction of the impulse and the direction of motion of \(P\) immediately before the impulse is applied. [3]

A particle of mass 0.3 kg is moving with velocity \((5\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\) when it receives an impulse \((-3\mathbf{i} + 3\mathbf{j})\) N s. Find the change in the kinetic energy of the particle due to the impulse. [6]

Three particles \(A\), \(B\) and \(C\) lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The masses of \(A\), \(B\) and \(C\) are \(3m\), \(4m\), and \(5m\) respectively. Particle \(A\) is projected with speed \(u\) towards particle \(B\) and collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{3}\).

1. Show that the impulse exerted by \(A\) on \(B\) in this collision has magnitude \(\frac{16}{7}mu\) [7]

After the collision between \(A\) and \(B\) there is a direct collision between \(B\) and \(C\). After this collision between \(B\) and \(C\), the kinetic energy of \(C\) is \(\frac{72}{245}mu^2\)

1. Find the coefficient of restitution between \(B\) and \(C\). [6]

The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) lie in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) vertical. A ball of mass \(0.1\) kg is hit by a bat which gives it an impulse of \((3.5\mathbf{i} + 3\mathbf{j})\) Ns. The velocity of the ball immediately after being hit is \((10\mathbf{i} + 25\mathbf{j})\) m s\(^{-1}\).

1. Find the velocity of the ball immediately before it is hit. [3]

In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.

1. Find the greatest height of the ball above the ground in the subsequent motion. [3]

The ball is caught when it is again 1 m above the ground.

1. Find the distance from the point where the ball is hit to the point where it is caught. [4]

\includegraphics{figure_3} A ball \(B\) of mass 0.4 kg is struck by a bat at a point \(O\) which is 1.2 m above horizontal ground. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are respectively horizontal and vertical. Immediately before being struck, \(B\) has velocity \((-20\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\). Immediately after being struck it has velocity \((15\mathbf{i} + 16\mathbf{j})\) m s\(^{-1}\). After \(B\) has been struck, it moves freely under gravity and strikes the ground at the point \(A\), as shown in Fig. 3. The ball is modelled as a particle.

1. Calculate the magnitude of the impulse exerted by the bat on \(B\). [4]
2. By using the principle of conservation of energy, or otherwise, find the speed of \(B\) when it reaches \(A\). [6]
3. Calculate the angle which the velocity of \(B\) makes with the ground when \(B\) reaches \(A\). [4]
4. State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic. [2]

A particle \(P\) of mass 0.4 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by $$\mathbf{r} = (t^2 + 4t)\mathbf{i} + (3t - t^3)\mathbf{j}.$$

1. Calculate the speed of \(P\) when \(t = 3\). [5]

When \(t = 3\), the particle \(P\) is given an impulse \((8\mathbf{i} - 12\mathbf{j})\) N s.

1. Find the velocity of \(P\) immediately after the impulse. [3]

A particle \(A\) of mass \(2m\) is moving with speed \(3u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2u\) in the opposite direction to \(A\). Immediately after the collision the speed of \(B\) is \(\frac{8}{3}u\) and the direction of motion of \(B\) is reversed.

1. Calculate the coefficient of restitution between \(A\) and \(B\). [6]
2. Show that the kinetic energy lost in the collision is \(7mu^2\). [3]

After the collision \(B\) strikes a fixed vertical wall that is perpendicular to the direction of motion of \(B\). The magnitude of the impulse of the wall on \(B\) is \(\frac{14}{3}mu\).

1. Calculate the coefficient of restitution between \(B\) and the wall. [4]