6.03a Linear momentum: p = mv

At a firing range, a man holds a gun and fires a bullet horizontally. The bullet is fired with a horizontal velocity of \(400 \text{ m s}^{-1}\). The mass of the gun is \(1.5\) kg and the mass of the bullet is \(30\) grams.

1. Find the speed of recoil of the gun. [2 marks]
2. Find the magnitude of the impulse exerted by the man on the gun in bringing the gun to rest after the bullet is fired. [2 marks]

\includegraphics{figure_2} A small ball \(Q\) of mass \(2m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac{13}{12}u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d_1\) from \(B\), and \(BC\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d_2\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(BC\), as shown in Fig. 2, where \(\tan \alpha = \frac{5}{12}\). The line of centres of \(P\) and \(Q\) is parallel to \(BC\). After the collision \(Q\) moves towards \(C\) with speed \(\frac{5}{4}u\).

1. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(CB\) are \(\frac{1}{4}u\) and \(\frac{5}{12}u\) respectively. [4]
2. Find the coefficient of restitution between \(P\) and \(Q\). [2]
3. Show that when \(Q\) reaches \(C\), \(P\) is at a distance \(\frac{4}{5}d_1\) from \(W\). [3]

For each collision between a ball and a wall the coefficient of restitution is \(\frac{1}{2}\). Given that the balls collide with each other again,

1. show that the time between the two collisions of the balls is \(\frac{15d_1}{u}\). [4]
2. find the ratio \(d_1 : d_2\). [5]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of \(A\) is 2.5 m s\(^{-1}\) and the speed of \(B\) is 1.3 m s\(^{-1}\). When they collide the line joining their centres makes an angle \(\alpha\) with the direction of motion of \(A\) and an angle \(\beta\) with the direction of motion of \(B\), where \(\tan \alpha = \frac{4}{3}\) and \(\tan \beta = \frac{12}{5}\) as shown in Fig. 1.

1. Find the components of the velocities of \(A\) and \(B\) perpendicular and parallel to the line of centres immediately before the collision. [4]

The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).

1. Find, to one decimal place, the speed of each sphere after the collision. [9]

\includegraphics{figure_3} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point \(X\) with a stationary van of mass 800 kg. After the collision the van came to rest at the point \(A\) having travelled a horizontal distance of 45 m, and the car came to rest at the point \(B\) having travelled a horizontal distance of 21 m. The angle \(AXB\) is 90°. The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.

1. Find the coefficient of restitution between the car and the van. [5]

The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along \(AX\) and the car to a constant horizontal force of 300 N along \(BX\).

1. Find the speed of the car immediately before the collision. [9]

[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]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors.] The vector \(\mathbf{n} = (-\frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j})\) and the vector \(\mathbf{p} = (-\frac{4}{5}\mathbf{i} + \frac{3}{5}\mathbf{j})\) are perpendicular unit vectors.

1. Verify that \(\frac{3}{5}\mathbf{n} + \frac{4}{5}\mathbf{p} = (\mathbf{i} + 3\mathbf{j})\). [2]

A smooth uniform sphere \(S\) of mass 0.5 kg is moving on a smooth horizontal plane when it collides with a fixed vertical wall which is parallel to \(\mathbf{p}\). Immediately after the collision the velocity of \(S\) is \((\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\). The coefficient of restitution between \(S\) and the wall is \(\frac{3}{5}\).

1. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the velocity of \(S\) immediately before the collision. [5]
2. Find the energy lost in the collision. [3]

Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(km\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of \(60°\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac{3u}{4(k + 1)}\). [6] Immediately after the collision the direction of motion of \(A\) makes an angle arctan \((2\sqrt{3})\) with the direction of motion of \(B\).
2. Show that \(k = \frac{1}{2}\). [6]
3. Find the loss of kinetic energy due to the collision. [4]

\includegraphics{figure_1} A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\). Show that \((1-e) \tan \beta = 2 \tan \alpha\). [11]

A bungee jumper of mass \(m\) kg is attached to an elastic rope. The other end of the rope is attached to a fixed point. The bungee jumper falls vertically from the fixed point. At time \(t\) seconds after the rope first becomes taut, the extension of the rope is \(x\) metres and the speed of the bungee jumper is \(v\) m s\(^{-1}\)

1. A model for the motion while the rope remains taut assumes that the forces acting on the bungee jumper are • the weight of the bungee jumper • a tension in the rope of magnitude \(kx\) newtons • an air resistance force of magnitude \(Rv\) newtons where \(k\) and \(R\) are constants such that \(4km > R^2\)
   1. Show that this model gives the result $$x = e^{-\frac{Rt}{2m}} \left( A \cos \frac{\sqrt{4km - R^2}}{2m} t + B \sin \frac{\sqrt{4km - R^2}}{2m} t \right) + \frac{mg}{k}$$ where \(A\) and \(B\) are constants, and \(g\) m s\(^{-2}\) is the acceleration due to gravity. You do not need to find the value of \(A\) or the value of \(B\) [6 marks]
   2. It is also given that: \(k = 16\) \(R = 20\) \(m = 62.5\) \(g = 9.8\) m s\(^{-2}\) and that the speed of the bungee jumper when the rope becomes taut is 14 m s\(^{-1}\) Show that, to the nearest integer, \(A = -38\) and \(B = 16\) [6 marks]
2. A second, simpler model assumes that the air resistance is zero. The values of \(k\), \(m\) and \(g\) remain the same. Find an expression for \(x\) in terms of \(t\) according to this simpler model, giving the values of all constants to two significant figures. [4 marks]

The position vector \(\mathbf{x}\) metres at time \(t\) seconds of an object of mass 3 kg may be modelled by $$\mathbf{x} = 3\sin t \mathbf{i} - 4\cos 2t \mathbf{j} + 5\sin t \mathbf{k}.$$

1. Find an expression for the velocity vector \(\mathbf{v}\text{ ms}^{-1}\) at time \(t\) seconds and determine the least value of \(t\) when the object is instantaneously at rest. [7]
2. Write down the momentum vector at time \(t\) seconds. [1]
3. Find, in vector form, an expression for the force acting on the object at time \(t\) seconds. [3]

Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \text{ ms}^{-1}\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics{figure_4}

1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(v_A \text{ ms}^{-1}\) and \(v_B \text{ ms}^{-1}\) respectively. \(\bullet\) Show that \(v_A = \frac{u(3-2e)}{5}\). \(\bullet\) Find an expression for \(v_B\) in terms of \(u\) and \(e\). [4]
2. Find an expression in terms of \(u\) and \(e\) for the velocity of \(B\) immediately after its collision with \(C\). [4]

After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).

1. Determine the range of possible values of \(e\). [4]