6.03f Impulse-momentum: relation

A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds, \(\mathbf{F} = (1.5t^2 - 3)\mathbf{i} + 2t\mathbf{j}\). When \(t = 2\), the velocity of \(P\) is \((-4\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\).

1. Find the acceleration of \(P\) at time \(t\) seconds. [2]
2. Show that, when \(t = 3\), the velocity of \(P\) is \((9\mathbf{i} + 15\mathbf{j})\) m s\(^{-1}\). [5]

When \(t = 3\), the particle \(P\) receives an impulse \(\mathbf{Q}\) N s. Immediately after the impulse the velocity of \(P\) is \((-3\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\). Find

1. the magnitude of \(\mathbf{Q}\), [3]
2. the angle between \(\mathbf{Q}\) and \(\mathbf{i}\). [3]

At time \(t\) seconds \((t \geq 0)\), a particle \(P\) has position vector \(\mathbf{p}\) metres, with respect to a fixed origin \(O\), where $$\mathbf{p} = (3t^2 - 6t + 4)\mathbf{i} + (3t^3 - 4t)\mathbf{j}.$$ Find

1. the velocity of \(P\) at time \(t\) seconds, [2]
2. the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf{i}\). [3]

When \(t = 1\), the particle \(P\) receives an impulse of \((2\mathbf{i} - 6\mathbf{j})\) N s. Given that the mass of \(P\) is 0.5 kg,

1. find the velocity of \(P\) immediately after the impulse. [4]

\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]

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]

A tennis ball of mass \(0.2\) kg is moving with velocity \((-10\mathbf{i})\) m s\(^{-1}\) when it is struck by a tennis racket. Immediately after being struck, the ball has velocity \((15\mathbf{i}+ 15\mathbf{j})\) m s\(^{-1}\). Find

1. the magnitude of the impulse exerted by the racket on the ball, [4]
2. the angle, to the nearest degree, between the vector \(\mathbf{i}\) and the impulse exerted by the racket, [2]
3. the kinetic energy gained by the ball as a result of being struck. [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]

[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]

Two smooth spheres \(A\) and \(B\), of masses 60 grams and 90 grams respectively, are at rest on a smooth horizontal table. \(A\) is projected towards \(B\) with speed 4 ms\(^{-1}\) and the particles collide. After the collision, \(A\) and \(B\) move in the same direction as each other, with speeds \(u\) ms\(^{-1}\) and \(6u\) ms\(^{-1}\) respectively. Calculate

1. the value of \(u\), [4 marks]
2. the magnitude of the impulse exerted by \(A\) on \(B\), stating the units of your answer. [3 marks]

\(A\) and \(B\) are now replaced in their original positions and projected towards each other with speeds 2 ms\(^{-1}\) and 8 ms\(^{-1}\) respectively. They collide again, after which \(A\) moves with speed 7 ms\(^{-1}\), its direction of motion being reversed.

1. Find the speed of \(B\) after this collision and state whether its direction of motion has been reversed. [5 marks]

Two railway trucks \(A\) and \(B\), of masses 10 000 kg and 7 000 kg respectively, are moving towards each other along a horizontal straight track. The trucks collide, and in the collision \(A\) exerts an impulse on \(B\) of magnitude 84 000 Ns. Immediately after the collision, the trucks move together with speed 10 ms\(^{-1}\). Modelling the trucks as particles,

1. find the speed of each truck immediately before the collision. [6 marks]

When the trucks are moving together along the track, the coefficient of friction between them and the track is 0.15. Assuming that no other resisting forces act on the trucks, calculate

1. the magnitude of the resisting force on the trucks, [3 marks]
2. the time taken after the collision for the trucks to come to rest. [3 marks]

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 2 kg and 1.5 kg respectively are at rest on a smooth, horizontal surface. They are connected by a light, inelastic string which is initially slack. Particle \(P\) is projected away from \(Q\) with a speed of 7 ms\(^{-1}\).

1. Find the common speed of the particles after the string becomes taut. [3 marks]
2. Calculate the impulse in the string when it jerks tight. [2 marks]

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]

In a safety test, a car of mass 800 kg is driven directly at a wall at a constant speed of 15 m s\(^{-1}\). The constant force exerted by the wall on the car in bringing it to rest is 60 kN.

1. Calculate the magnitude of the impulse exerted by the wall on the car. [2 marks]
2. Find the time it takes for the car to come to rest. [2 marks]
3. Show that the deceleration of the car is 75 m s\(^{-2}\). [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]

Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(3m\) respectively, are moving on a smooth horizontal table with velocities \((3\mathbf{i} - \mathbf{j})\) ms\(^{-1}\) and \((4\mathbf{i} + \mathbf{j})\) ms\(^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. They collide, after which \(A\) has velocity \((5\mathbf{i} + \mathbf{j})\) ms\(^{-1}\).

1. Find the magnitude of the impulse exerted on \(B\) by \(A\), stating the units of your answer. [4 marks]
2. Find the speed of \(B\) immediately after the collision. [5 marks]

A ball, of mass \(m\) kg, is moving with velocity \((5\mathbf{i} - 3\mathbf{j})\) ms\(^{-1}\) when it receives an impulse of \((-2\mathbf{i} - 4\mathbf{j})\) Ns. Immediately after the impulse is applied, the ball has velocity \((3\mathbf{i} + k\mathbf{j})\) ms\(^{-1}\). Find the values of the constants \(k\) and \(m\). [6 marks]

\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]