Momentum

An electric car of mass 1700 kg is tested to measure its maximum speed.

Fig. 16.1 shows a graph of velocity v against time t for the first 18 seconds of the test.

During this time the force produced by the motor is constant.

Fig. 16.1

Describe using Fig. 16.1, the method you would use to estimate the distance travelled in 18 s.

Use the graph in Fig. 16.1 to show that the acceleration of the car at time t = 12 s is about 1.5 m s^–2.

Use Fig. 16.1 to calculate the change of momentum of the car in 18 s.

=........ kg m s^–1

Fig. 16.2 shows a graph of resultant force against time during the test.

Fig. 16.2

Explain, in terms of the forces acting on the car, the shape of the graph in Fig. 16.2.

Explain why the value you have calculated in (a)(iii) is the area under the graph in Fig. 16.2.

The diagram below shows one of the wheels of the car that is driven by the electric motor.

Use Newton’s third law to explain how frictional forces cause the car to accelerate.

Calculate the resultant force acting on the car at t = 12 s, using your answer to (a)(ii).

A sealed container contains n moles of an ideal gas. The gas has pressure p, absolute temperature T and occupies volume V.

The mass of one mole of the gas is M.

Use an ideal gas equation to show that the density of the gas is given by the expression

An airship has a cabin suspended underneath a gasbag inflated with helium.

The airship is floating above the ground and is stationary.

The volume of the gasbag is 12 000 m³. The temperature of the helium and the surrounding air is 20 °C. Atmospheric pressure is 1.0 × 10⁵ Pa. The molar mass of air is 0.029 kg mol^–1. The volume of the cabin is negligible compared to the volume of the gasbag.

Show that the density of air under the conditions described is about 1.2 kg m^–3. (i)

Calculate the weight of air displaced by the airship.

weight of air........ N

Explain why the weight of air displaced by the airship has the same magnitude as the weight of the airship and its contents.

The pressure of the helium in the gasbag is maintained at a value only slightly greater than atmospheric pressure. Suggest why a larger pressure is not used.

The airship engine drives a fan which moves 7.8 kg of air per second at a relative speed of 45 m s^–1, so the airship starts to move.

All other conditions given in (b) remain the same.

Calculate the thrust that the engine produces.

The airship has a higher maximum speed at high altitudes, but also produces less thrust from the engine.

Explain these observations.

An archer fires an arrow towards a target as shown below.

The diagram is not drawn to scale.

The centre of the target is at the same height as the initial position of the arrow. The target is a distance of 90 m from the arrow. The arrow has an initial velocity of 68 m s^–1 and is fired at an angle of 11 to the horizontal.

Air resistance has negligible effect on the motion of the arrow.

Describe how the kinetic energy of the arrow changes during its journey from when it is fired until it reaches its maximum height.

Show that the time taken for the arrow to reach its maximum height is about 1.3 s.

The arrow misses the target.

Calculate the horizontal distance, measured along the base line, by which the arrow misses the target.

Explain what is meant by an inelastic collision.

The target is mounted on wheels. The target has a much larger mass than the mass of the arrow.

Using ideas of momentum, explain the velocity of the target immediately after the arrow sticks into the target.

A kiln used to harden ceramics is shown below.

The internal chamber is a cube. Each side of this cube has length 0.46 m. The chamber is sealed and full of argon. Argon behaves as an ideal gas.

The kiln is initially at 20 C. The argon in the kiln has an initial pressure of 100 kPa.

Calculate the amount of argon n in the chamber in moles.

The temperature of the kiln is increased from 20 C to 1300 C.

Calculate the pressure in kPa at 1300 C.

The temperature of the kiln is 1300 C. A single atom of argon is travelling horizontally towards the vertical side X of the chamber. The initial speed of this atom is 990 m s^–1. After collision, it rebounds at the same speed.

Calculate the change in momentum p of this atom.

• mass of argon atom = 6.6 × 10^–26 kg

=........ kg m s^–1 p

Assume that this atom does not collide with any other argon atoms inside the chamber. Instead, it travels horizontally, making repeated collisions with the opposite vertical walls of the chamber.

1 Show that the atom makes about 1000 collisions with side X in a time interval of 1.0 s.

2 Calculate the average force F on side X made by the atom.

Without calculation, explain how your answer to (ii)2 could be used to estimate the total pressure exerted by the atoms of the argon gas in the kiln.

A beam of -particles is incident on a thin gold foil. Most -particles pass straight through the foil. A few are deflected by gold nuclei.

The diagram shows the path of one -particle which passes close to a gold nucleus N in the foil. The -particle is deflected through an angle of 60° as it travels from A to B.

P marks its position of closest approach to the gold nucleus.

Another -particle in the beam is deflected by the same gold nucleus N through an angle of 30°.

Sketch its path onto the diagram above.

The initial kinetic energy of each -particle is 5.0 MeV.

Show that the magnitude of the initial momentum of each -particle is about 10^–19 kg m s^–1 Take the mass of the -particle to be 6.6 × 10^–27 kg.

Draw two vectors in the spaces below to represent the initial momentum and the final momentum of the -particle.

initial momentum at A

final momentum at B

Draw a vector in the space below to represent the momentum of the nucleus N when the -particle reaches B.

Explain how you determined this momentum.

A bicycle manufacturer carries out tests on the braking system of their new model. A cyclist on this new bicycle travels at a constant initial speed U. The cyclist applies the brakes at time t = 0 and the bicycle comes to a stop at time t = 2.0 s.

Fig. 20.1 shows the variation of the braking force F on the bicycle with time t.

Use Newton’s second law of motion to explain the physical quantity represented by the area under the graph shown in Fig. 20.1.

The total mass of cyclist and bicycle is 71 kg.

Use Fig. 20.1 to calculate the initial speed U.

=........ m s^–1 U

Complete Fig. 20.2 to show the variation of the speed of the bicycle from t = 0 to t = 2.0 s.

A puck of mass 0.16 kg is sliding on ice with a constant velocity of 11.0 m s⁻¹. A hockey stick exerts 13 a force on the puck, for a short period of time, in the opposite direction to the velocity of the puck. The momentum of the puck changes by 2.0 kg m s⁻¹.

Ignore friction.

What is the speed of the puck when it leaves the hockey stick?

1.5 m s⁻¹ A

3.8 m s⁻¹ B

12.5 m s⁻¹ C

23.5 m s⁻¹ D

Your answer