6.03e Impulse: by a force

\includegraphics{figure_1} The points \(A\), \(B\) and \(C\) lie in a horizontal plane. A batsman strikes a ball of mass \(0.25\) kg. Immediately before being struck, the ball is moving along the horizontal line \(AB\) with speed \(30 \text{ ms}^{-1}\). Immediately after being struck, the ball moves along the horizontal line \(BC\) with speed \(40 \text{ ms}^{-1}\). The line \(BC\) makes an angle of \(60°\) with the original direction of motion \(AB\), as shown in Figure 1. Find, to 3 significant figures,

1. the magnitude of the impulse given to the ball,
2. the size of the angle that the direction of this impulse makes with the original direction of motion \(AB\).

[8]

A tennis ball of mass \(0.1\) kg is hit by a racquet. Immediately before being hit, the ball has velocity \(30\mathbf{i}\) m s\(^{-1}\). The racquet exerts an impulse of \((-2\mathbf{i} - 4\mathbf{j})\) N s on the ball. By modelling the ball as a particle, find the velocity of the ball immediately after being hit. [4]

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

1. the magnitude of the impulse of the bat on the ball, (4)
2. the size of the angle between the vector \(\mathbf{i}\) and the impulse exerted by the bat on the ball, (2)
3. the kinetic energy lost by the ball in the impact. (3)

A ball of mass 0.5 kg is moving with velocity \(12\mathbf{i}\) m s\(^{-1}\) when it is struck by a bat. The impulse received by the ball is \((-4\mathbf{i} + 7\mathbf{j})\) N s. By modelling the ball as a particle, find

1. the speed of the ball immediately after the impact, [4]
2. the angle, in degrees, between the velocity of the ball immediately after the impact and the vector \(\mathbf{i}\), [2]
3. the kinetic energy gained by the ball as a result of the impact. [2]

A tennis ball, moving horizontally, hits a wall at \(25 \text{ ms}^{-1}\) and rebounds along the same straight line at \(15 \text{ ms}^{-1}\). The impulse exerted by the wall on the ball has magnitude \(12\) Ns.

1. Calculate the mass of the ball. [4 marks]
2. State any modelling assumptions that you have made. [2 marks]

Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(m\) respectively, are connected by a light inextensible string which passes over a smooth fixed pulley as shown. \(A\) is initially at rest on the rough horizontal surface of a table, the coefficient of friction between \(A\) and the table being \(\frac{2}{7}\). \(B\) hangs freely on the end of the vertical portion of the string. \includegraphics{figure_5} \(A\) is now given an impulse, directed away from the pulley, of magnitude \(5m\) Ns.

1. Show that the system starts to move with speed \(2.5 \text{ ms}^{-1}\). [1 mark]
2. State which modelling assumption ensures that the tensions in the two sections of the string can be taken to be equal. [1 mark]

Given that \(A\) comes to rest before it reaches the edge of the table and before \(B\) hits the pulley,

1. find the time taken for the system to come to rest. [7 marks]
2. Find the distance travelled by \(A\) before it first comes to rest. [4 marks]

\(A\), \(B\) and \(C\) are three small spheres of equal radii and masses \(2m\), \(m\) and \(5m\) respectively. They are placed in a straight line on a smooth horizontal surface. \(A\) is projected with speed 6 ms\(^{-1}\) towards \(B\), which is at rest. When \(A\) hits \(B\) it exerts an impulse of magnitude \(8m\) Ns on \(B\).

1. Find the speed with which \(B\) starts to move. [2 marks]
2. Show that the speed of \(A\) after it collides with \(B\) is 2 ms\(^{-1}\). [3 marks]

After travelling 3 m, \(B\) hits \(C\), which is then travelling towards \(B\) at \(2.2\) ms\(^{-1}\). \(C\) is brought to rest by this impact.

1. Show that the direction of \(B\)'s motion is reversed and find its new speed. [3 marks]
2. Find how far \(B\) now travels before it collides with \(A\) again. [6 marks]
3. State a modelling assumption that you have made about the spheres. [1 mark]

Two small smooth spheres \(A\) and \(B\), of equal radius but masses \(m\) kg and \(km\) kg respectively, where \(k > 1\), move towards each other along a straight line and collide directly. Immediately before the collision, \(A\) has speed 5 ms\(^{-1}\) and \(B\) has speed 3 ms\(^{-1}\). In the collision, the impulse exerted by \(A\) on \(B\) has magnitude \(7km\) Ns.

1. Find the speed of \(B\) after the impact. [3 marks]
2. Show that the speed of \(A\) immediately after the collision is \((7k - 5)\) ms\(^{-1}\) and deduce that the direction of \(A\)'s motion is reversed. [5 marks]

\(B\) is now given a further impulse of magnitude \(mu\) Ns, as a result of which a second collision between it and \(A\) occurs.

1. Show that \(u > k(7k - 1)\). [4 marks]

Two trucks \(P\) and \(Q\), of masses 18 000 kg and 16 000 kg respectively, collide while moving towards each other in a straight line. Immediately before the collision, both trucks are travelling at the same speed, \(u\) ms\(^{-1}\). Immediately after the collision, \(P\) is moving at half its original speed, its direction of motion having been reversed.

1. Find, in terms of \(u\), the speed of \(Q\) immediately after the collision. \hfill [5 marks]
2. State, with a reason, whether the direction of \(Q\)'s motion has been reversed. \hfill [1 mark]
3. Find, in terms of \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision, stating the units of your answer. \hfill [3 marks]

The force exerted by each truck on the other in the impact has magnitude \(108000u\) N.

1. Find the time for which the trucks are in contact. \hfill [3 marks]

Two particles \(P\) and \(Q\), of mass \(m\) and \(km\) respectively, are travelling in opposite directions on a straight horizontal path with speeds \(3u\) and \(2u\) respectively. \(P\) and \(Q\) collide and, as a result, the direction of motion of both particles is reversed and their speeds are halved.

1. Find the value of \(k\). [4 marks]
2. Write down an expression in terms of \(m\) and \(u\) for the magnitude of the impulse which \(P\) exerts on \(Q\) during the collision. [3 marks]

\includegraphics{figure_4} Three smooth spheres \(A\), \(B\) and \(C\), of equal radius and of masses \(m\) kg, \(2m\) kg and \(3m\) kg respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed 5 m s\(^{-1}\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed 2 m s\(^{-1}\).

1. Find the coefficient of restitution between \(A\) and \(B\). [4]
2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. [2]

Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.

1. Show that there will be another collision. [3]

\includegraphics{figure_4} A uniform square lamina \(ABCD\) of side 6 cm has a semicircular piece, with \(AB\) as diameter, removed (see diagram).

1. Find the distance of the centre of mass of the remaining shape from \(CD\). [6]

The remaining shape is suspended from a fixed point by a string attached at \(C\) and hangs in equilibrium.

1. Find the angle between \(CD\) and the vertical. [2]

A particle of mass 0.5 kg is held at rest at a point \(P\), which is at the bottom of an inclined plane. The particle is given an impulse of 1.8 N s directed up a line of greatest slope of the plane.

1. Find the speed at which the particle starts to move. [2]

The particle subsequently moves up the plane to a point \(Q\), which is 0.3 m above the level of \(P\).

1. Given that the plane is smooth, find the speed of the particle at \(Q\). [4]

It is given instead that the plane is rough. The particle is now projected up the plane from \(P\) with initial speed 3 ms\(^{-1}\), and comes to rest at a point \(R\) which is 0.2 m above the level of \(P\).

1. Given that the plane is inclined at 30° to the horizontal, find the magnitude of the frictional force on the particle. [4]

A particle \(P\) is projected with speed \(32 \text{ m s}^{-1}\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac{3}{4}\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac{24}{25}\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).

1. Calculate the height of \(C\) above the ground and the distance \(AB\). [4]

Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.

1. Given that the mass of \(P\) is 3 kg, find the magnitude and direction of the impulse exerted on \(P\) by the ground. [4]

The coefficient of restitution between the two particles is \(\frac{1}{2}\).

1. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25°\) below the horizontal. [9]

A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \text{ m s}^{-1}\) and \(6 \text{ m s}^{-1}\) respectively. The speed of the sphere immediately after it bounces is \(5 \text{ m s}^{-1}\).

1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \text{ m s}^{-1}\), and hence find the coefficient of restitution between the surface and the sphere. [3]
2. State the direction of the impulse on the sphere and find its magnitude. [3]

A small ball of mass \(0.8\) kg is moving with speed \(10.5\) m s\(^{-1}\) when it receives an impulse of magnitude \(4\) N s. The speed of the ball immediately afterwards is \(8.5\) m s\(^{-1}\). The angle between the directions of motion before and after the impulse acts is \(\alpha\). Using an impulse-momentum triangle, or otherwise, find \(\alpha\). [6]

\includegraphics{figure_1} A particle \(P\) of mass \(0.3\) kg is moving in a straight line with speed \(4\) m s\(^{-1}\) when it is deflected through an angle \(\theta\) by an impulse of magnitude \(I\) N s. The impulse acts at right angles to the initial direction of motion of \(P\) (see diagram). The speed of \(P\) immediately after the impulse acts is \(5\) m s\(^{-1}\). Show that \(\cos \theta = 0.8\) and find the value of \(I\). [4]

A particle \(P\) of mass \(0.2\) kg is moving on a smooth horizontal surface with speed \(3\text{ ms}^{-1}\), when it is struck by an impulse of magnitude \(I\) Ns. The impulse acts horizontally in a direction perpendicular to the original direction of motion of \(P\), and causes the direction of motion of \(P\) to change by an angle \(\alpha\), where \(\tan \alpha = \frac{5}{12}\).

1. Show that \(I = 0.25\). [4]
2. Find the speed of \(P\) after the impulse acts. [2]

\includegraphics{figure_1} A particle \(P\) of mass \(0.3\) kg is moving with speed \(0.4\) m s\(^{-1}\) in a straight line on a smooth horizontal surface when it is struck by a horizontal impulse. After the impulse acts \(P\) has speed \(0.6\) m s\(^{-1}\) and is moving in a direction making an angle \(30°\) with its original direction of motion (see diagram).

1. Find the magnitude of the impulse and the angle its line of action makes with the original direction of motion of \(P\). [4]

Subsequently a second impulse acts on \(P\). After this second impulse acts, \(P\) again moves from left to right with speed \(0.4\) m s\(^{-1}\) in a direction parallel to its original direction of motion.

1. State the magnitude of the second impulse, and show the direction of the second impulse on a diagram. [2]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors.] Two smooth uniform spheres \(A\) and \(B\) have equal radius but masses \(m\) and \(5m\) respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocities of \(A\) and \(B\) are \((\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and \((-\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\) respectively. Immediately after the collision, the velocity of \(A\) is \((-2\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\).

1. By considering the impulse on \(A\), find a unit vector parallel to the line joining the centres of the spheres when they collide. [4]
2. Find the velocity of \(B\) immediately after the collision. [3]

A small smooth ball of mass \(\frac{1}{2}\) kg is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{1}{3}\). Immediately before striking the plane the ball has speed 10 m s\(^{-1}\). The coefficient of restitution between ball and plane is \(\frac{1}{2}\). Find

1. the speed, to 3 significant figures, of the ball immediately after the impact, [5]
2. the magnitude of the impulse received by the ball as it strikes the plane. [2]

As a particle moves along a straight horizontal line, it is subjected to a force \(F\) newtons that acts in the direction of motion of the particle. At time \(t\) seconds, \(F = \frac{t}{5}\) Calculate the magnitude of the impulse on the particle between \(t = 0\) and \(t = 3\) Circle your answer. [1 mark] 0.3 N s \quad 0.6 N s \quad 0.9 N s \quad 1.8 N s

A light inextensible string passes over a smooth fixed pulley. Particles of mass 0.2 kg and 0.3 kg are attached to opposite ends of the string, so that the parts of the string not in contact with the pulley are vertical. The system is released from rest with the string taut.

1. Find the acceleration of the particles and the tension in the string. [6]

When the heavier particle has fallen 2.25 m it hits the ground and is brought to rest (and the string goes slack).

1. Find the speed with which it hits the ground. [2]
2. Find the magnitude of the impulse of the ground on the particle. [2]
3. If the impact between the particle and the ground lasts for 0.005 seconds, find the constant force that would be needed to bring the particle to rest. [2]

1. Determine the impulse of a force of magnitude \(6\) N that acts on a particle of mass \(3\) kg for \(1.5\) seconds. [1]

Particles \(A\) and \(B\), of masses \(0.1\) kg and \(0.2\) kg respectively, can move on a smooth horizontal table. Initially \(A\) is moving with speed \(3\) m s\(^{-1}\) towards \(B\), which is moving with speed \(1\) m s\(^{-1}\) in the same direction as the motion of \(A\). During a collision \(B\) experiences an impulse from \(A\) of magnitude \(0.2\) kg m s\(^{-1}\).

1. Find the speeds of the particles immediately after the collision. [4]
2. Determine the coefficient of restitution between the particles. [2]