6.02d Mechanical energy: KE and PE concepts

1 A particle \(Q\) of mass \(m \mathrm {~kg}\) is acted on by a single force so that it moves with constant acceleration \(\mathbf { a } = \binom { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\). Initially \(Q\) is at the point \(O\) and is moving with velocity \(\mathbf { u } = \binom { 2 } { - 5 } \mathrm {~ms} ^ { - 1 }\). After \(Q\) has been moving for 5 seconds it reaches the point \(A\).

1. Use the equation \(\mathbf { v } . \mathbf { v } = \mathbf { u } . \mathbf { u } + 2 \mathbf { a } . \mathbf { x }\) to show that at \(A\) the kinetic energy of \(Q\) is 37 m J .
2. 1. Show that the power initially generated by the force is - 8 mW W.
   2. The power in part (b)(i) is negative. Explain what this means about the initial motion of \(Q\).
3. 1. Find the time at which the power generated by the force is instantaneously zero.
   2. Find the minimum kinetic energy of \(Q\) in terms of \(m\).

1. A particle \(Q\) of mass 3 kg is moving on a smooth horizontal surface.

Particle \(Q\) is moving with velocity \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives a horizontal impulse of magnitude \(3 \sqrt { 82 } \mathrm { Ns }\). Immediately after receiving the impulse, the velocity of \(Q\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(x\) and \(y\) are positive constants. The kinetic energy gained by \(Q\) as a result of receiving the impulse is 138 J .
Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(Q\) immediately after receiving the impulse.

7 \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_339_511_349_817} A particle of mass 0.3 kg is attached to one end \(A\) of a light inextensible string of length 1.5 m . The other end \(B\) of the string is attached to a ceiling, so that the particle may swing in a vertical plane. The particle is released from rest when the string is taut and makes an angle of \(75 ^ { \circ }\) with the vertical (see diagram). Air resistance may be regarded as being negligible.

1. Show that, at an instant when the string makes an angle of \(40 ^ { \circ }\) with the vertical, the speed of the particle is \(3.90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
2. By considering Newton's second law, along and perpendicular to the string, find the radial and transverse components of acceleration, at this same instant, and hence the magnitude of the acceleration of the particle. \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_419_604_1370_772} A smooth sphere of mass 0.3 kg is moving in a straight line on a horizontal surface. It collides with a vertical wall when the velocity of the sphere is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the wall (see diagram). The coefficient of restitution between the sphere and the wall is 0.4 .
3. (a) Find the component of the velocity of the sphere perpendicular to the wall immediately after the collision.
   (b) Find the magnitude of the impulse exerted by the wall on the sphere.
4. Determine the magnitude and direction of the velocity of the sphere immediately after the collision, giving the direction as an acute angle to the wall.

7 A child of mass 20 kg slides down a rough slope of length 16 m against a constant frictional force \(F \mathrm {~N}\). Starting with an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at a point 8 m above the ground, she reaches the ground with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(F\).

Two particles, of masses \(3m\) and \(m\), are moving in the same straight line towards each other with speeds \(2u\) and \(u\) respectively. When they collide, the impulse acting on each particle has magnitude \(4mu\). Show that the total loss in kinetic energy is \(\frac{4}{5}mu^2\). [6]

Two particles, of masses \(3m\) and \(m\), are moving in the same straight line towards each other with speeds \(2u\) and \(u\) respectively. When they collide, the impulse acting on each particle has magnitude \(4mu\). Show that the total loss in kinetic energy is \(\frac{5}{2}mu^2\). [6]

Small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(10 \text{ ms}^{-1}\). After the spheres collide \(A\) continues to move in the same direction but with half the speed of \(B\).

1. Find the speed of \(B\) after the collision. [2]

A third small smooth sphere \(C\), of mass 1 kg and with the same radius as \(A\) and \(B\), is at rest on the plane. \(B\) now collides directly with \(C\). After this collision \(B\) continues to move in the same direction but with one third the speed of \(C\).

1. Show that there is another collision between \(A\) and \(B\). [3]
2. \(A\) and \(B\) coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions. [5]

Small smooth spheres \(A\) and \(B\), of equal radii and of masses \(5\text{kg}\) and \(3\text{kg}\) respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(8.5\text{ms}^{-1}\). The spheres collide and after the collision \(A\) continues to move in the same direction but with a quarter of the speed of \(B\).

1. Find the speed of \(B\) after the collision. [3]
2. Find the loss of kinetic energy of the system due to the collision. [2]

A particle \(P\) of mass \(0.4\) kg is projected vertically upwards from horizontal ground with speed \(10\) m s\(^{-1}\).

1. Find the greatest height above the ground reached by \(P\). [2]

When \(P\) reaches the ground again, it bounces vertically upwards. At the first instant that it hits the ground, \(P\) loses \(7.2\) J of energy.

1. Find the time between the first and second instants at which \(P\) hits the ground. [4]

A particle \(B\) of mass 5 kg is at rest on a smooth horizontal table. A particle \(A\) of mass 2.5 kg moves on the table with a speed of \(6 \text{ m s}^{-1}\) and collides directly with \(B\). In the collision the two particles coalesce.

1. Find the speed of the combined particle after the collision. [2]
2. Find the loss of kinetic energy of the system due to the collision. [3]

Small smooth spheres \(A\) and \(B\), of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(A\) is moving towards \(B\) with speed 5 m s\(^{-1}\) and \(B\) is moving towards \(A\) with speed 3 m s\(^{-1}\). After the spheres collide, both \(A\) and \(B\) move in the same direction and the difference in the speeds of the spheres is 2 m s\(^{-1}\). Find the loss of kinetic energy of the system due to the collision. [5]

A cyclist is riding a bicycle along a straight horizontal road \(AB\) of length 50 m. The cyclist starts from rest at \(A\) and reaches a speed of \(6 \text{ m s}^{-1}\) at \(B\). The cyclist produces a constant driving force of magnitude 100 N. There is a resistance force, and the work done against the resistance force from \(A\) to \(B\) is 3560 J. Find the total mass of the cyclist and bicycle. [3]

\includegraphics{figure_5} A cyclist and his machine have a total mass of 80 kg. The cyclist starts from rest at the top \(A\) of a straight path and freewheels (moves without pedalling or braking) down the path to \(B\). The path \(AB\) is inclined at 2.6° to the horizontal and is of length 250 m (see diagram).

1. Given that the cyclist passes through \(B\) with speed 9 m s\(^{-1}\), find the gain in kinetic energy and the loss in potential energy of the cyclist and his machine. Hence find the work done against the resistance to motion of the cyclist and his machine. [3]

The cyclist continues to freewheel along a horizontal straight path \(BD\) until he reaches the point \(C\), where the distance \(BC\) is \(d\) m. His speed at \(C\) is 5 m s\(^{-1}\). The resistance to motion is constant, and is the same on \(BD\) as on \(AB\).

1. Find the value of \(d\). [3]

The cyclist starts to pedal at \(C\), generating 425 W of power.

1. Find the acceleration of the cyclist immediately after passing through \(C\). [3]

A particle of mass \(0.4\) kg is projected with a speed of \(12\) m s\(^{-1}\) up a line of greatest slope of a smooth plane inclined at \(30°\) to the horizontal.

1. Find the initial kinetic energy of the particle. [1]
2. Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest. [3]

The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her speed at the bottom of the hill is 10 m s\(^{-1}\) and at the top it is 5 m s\(^{-1}\). There is a resistance to motion, and the work done against this resistance as the cyclist ascends the hill is 2000 J. The cyclist exerts a constant force of magnitude \(F\) N in the direction of motion. Find \(F\). [5]

\includegraphics{figure_6} Two uniform smooth spheres A and B of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle \(\alpha\) with the line of centres, and B's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{1}{3}\) and \(2\cos\beta = \cos\alpha\).

1. Show that the direction of motion of A after the collision is perpendicular to the line of centres. [4]

The total kinetic energy of the spheres after the collision is \(\frac{3}{4}mu^2\).

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

70% of the total kinetic energy of the spheres is lost as a result of the collision.

1. Given that \(\tan \theta = \frac{1}{3}\), find the value of \(k\). [6]

A brick of mass 3 kg slides in a straight line on a horizontal floor. The brick is modelled as a particle and the floor as a rough plane. The initial speed of the brick is 8 m s\(^{-1}\). The brick is brought to rest after moving 12 m by the constant frictional force between the brick and the floor.

1. Calculate the kinetic energy lost by the brick in coming to rest, stating the units of your answer. [2]
2. Calculate the coefficient of friction between the brick and the floor. [4]

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]

A particle of mass 0.8 kg is moving in a straight line on a rough horizontal plane. The speed of the particle is reduced from 15 m s\(^{-1}\) to 10 m s\(^{-1}\) as the particle moves 20 m. Assuming that the only resistance to motion is the friction between the particle and the plane, find

1. the work done by friction in reducing the speed of the particle from 15 m s\(^{-1}\) to 10 m s\(^{-1}\), [2]
2. the coefficient of friction between the particle and the plane. [4]

A parcel of mass 2.5 kg is moving in a straight line on a smooth horizontal floor. Initially the parcel is moving with speed 8 m s\(^{-1}\). The parcel is brought to rest in a distance of 20 m by a constant horizontal force of magnitude \(R\) newtons. Modelling the parcel as a particle, find

1. the kinetic energy lost by the parcel in coming to rest, [2]
2. the value of \(R\). [3]

A particle \(P\) of mass \(2m\) is moving with speed \(2u\) in a straight line on a smooth horizontal plane. A particle \(Q\) of mass \(3m\) is moving with speed \(u\) in the same direction as \(P\). The particles collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac{3}{2}u\). [5]
2. Find the total kinetic energy lost in the collision. [5]

After the collision between \(P\) and \(Q\), the particle \(Q\) collides directly with a particle \(R\) of mass \(m\) which is at rest on the plane. The coefficient of restitution between \(Q\) and \(R\) is \(e\).

1. Calculate the range of values of \(e\) for which there will be a second collision between \(P\) and \(Q\). [7]

A particle of mass \(0.5\) kg is projected vertically upwards from ground level with a speed of \(20 \text{ ms}^{-1}\). It comes to instantaneous rest at a height of \(10\) m above the ground. As the particle moves it is subject to air resistance of constant magnitude \(R\) newtons. Using the work-energy principle, or otherwise, find the value of \(R\). [6]

Three identical particles, \(A\), \(B\) and \(C\), lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The mass of each particle is \(m\). Particle \(A\) is projected towards \(B\) with speed \(u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac{2}{3}\).

1. Find, in terms of \(u\),
   1. the speed of \(A\) after this collision,
   2. the speed of \(B\) after this collision.
   [7]
2. Show that the kinetic energy lost in this collision is \(\frac{5}{36}mu^2\) [4]

After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\).

1. Find, in terms of \(u\), the speed of \(C\) immediately after this collision between \(B\) and \(C\). [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]