Direct collision, find impulse magnitude

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]

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 \(B\), 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]

\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 cricket ball of mass 0.5 kg is struck by a bat. Immediately before being struck, the velocity of the ball is \((-30\mathbf{i})\) m s\(^{-1}\). Immediately after being struck, the velocity of the ball is \((16\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\).

1. Find the magnitude of the impulse exerted on the ball by the bat. [4]

In the subsequent motion, the position vector of the ball is \(\mathbf{r}\) metres at time \(t\) seconds. In a model of the situation, it is assumed that \(\mathbf{r} = [16t\mathbf{i} + (20t - 5t^2)\mathbf{j}]\). Using this model,

1. find the speed of the ball when \(t = 3\). [4]

A particle \(A\) of mass \(3m\) is moving along a straight line with constant speed \(u\) m s\(^{-1}\). It collides with a particle \(B\) of mass \(2m\) moving at the same speed but in the opposite direction. As a result of the collision, \(A\) is brought to rest.

1. Show that, after the collision, \(B\) has changed its direction of motion and that its speed has been halved. [4 marks]

Given that the magnitude of the impulse exerted by \(A\) on \(B\) is \(9m\) Ns,

1. find the value of \(u\). [3 marks]

A particle, \(P\), of mass 5 kg moves with speed 3 m s\(^{-1}\) along a smooth horizontal track. It strikes a particle \(Q\) of mass 2 kg which is at rest on the track. Immediately after the collision, \(P\) and \(Q\) move in the same direction with speeds \(v\) and 2v m s\(^{-1}\) respectively.

1. Calculate the value of \(v\). [3 marks]
2. Calculate the magnitude of the impulse received by \(Q\) on impact. [2 marks]