Acceleration

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).

An aircraft is flying horizontally at an altitude of 10 000 m and a constant speed of 220 m s^–1. 17

While maintaining this speed, the controls are adjusted so that the aircraft is in projectile motion until the altitude is 7500 m.

Throughout the projectile motion the passengers experience apparent weightlessness.

A condition for projectile motion is constant vertical acceleration. State the other condition.

Show that t, the time taken to reach the new altitude of 7500 m, is about 23 s.

Determine the magnitude and direction of the velocity of the aircraft at time t.

velocity =........ m s

angle to the vertical........

Explain why the engines of the aircraft must provide thrust during the projectile motion.

Explain why the passengers are described as ‘weightless’ during the projectile motion of the aircraft.

Draw an arrow labelled T on the diagram to represent the tension in the rod.

Show that the radius of the circular path followed by the sandbag is about 10 m.

Calculate the tension T in the rod.

Show that the angular velocity of the ride is about 0.8 radians per second.

When the seat is at its highest point the sandbag is 17 m above the ground. The sandbag is released from the seat to model an object being dropped by a rider.

Calculate t, the time taken for the sandbag to reach the ground.

Using your answer to (a)(iv), determine the horizontal displacement s travelled by the sandbag before hitting the ground.

Determine, with reasons, the effect on the horizontal displacement travelled if the object released from the ride was a shoe from a rider.

18* A student is attempting to determine the value of g, the acceleration due to gravity, by two different methods.

Method 1. Vertical drop method

Measuring the time of fall of a small dense ball being dropped from rest from different heights.

Method 2. Rolling ball method

Measuring the time it takes for the same ball to roll 1.900 m down a ramp, set at different angles.

Single sets of results are shown below. The times were measured using a standard stopwatch operated by the student.

Compare and discuss the uncertainties of the two values of g that could be obtained using these single measurements.

Describe how the student would analyse both sets of data when a full range of results has been taken.

Additional space if required

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.

A cloud is made up of droplets of water falling at terminal velocity.

Describe and explain the motion of an object falling at terminal velocity.

The terminal velocity v of a small sphere of density _s and radius r falling through a fluid of density _f is given by the formula:

where is a constant for the fluid and g is the acceleration of free fall.

Water droplets of rain fall to the ground whereas water droplets in mist appear to float.

Use the formula above to suggest why.

A tennis ball is hit with a racket. The force applied by the racket on the ball is F. The ball has a vertical path through the air.

Which statement is correct when the ball is at its maximum height?

A The ball has a downward acceleration.

B The force acting on the ball is F.

C The ball experiences greatest drag.

D The weight of the ball is equal to the drag.

Your answer

The velocity v against time t graphs for four objects A, B, C and D are shown below.

Which object travels the greatest distance between t = 0 and t = T ?

Your answer

A paper cone is held above the ground and dropped. It falls vertically and reaches terminal velocity before it hits the ground.

Which statement correctly describes the resultant force on the falling cone before it reaches terminal velocity?

A decreasing and upwards

B decreasing and downwards

C increasing and downwards

D increasing and upwards

Your answer

A student uses a motion-sensor connected to a laptop to investigate the motion of a hollow ball of mass 1.2 × 10^–2 kg falling through air.

The ball is dropped from rest. It reaches terminal velocity before it reaches the ground.

The upthrust on the ball is negligible.

Fig. 17 shows the variation with time t of the velocity v of the ball as it falls towards the ground.

Draw a tangent to the curve at t = 0.25 s and determine the acceleration of the ball.

acceleration =........ m s^–2

Calculate the resultant force F acting on the ball at t = 0.25 s.

Use your answer in (b) to calculate the drag on the ball at time t = 0.25 s.

The student now adds a small amount of sand inside the hollow ball. As before, the ball is dropped from rest and it also reaches terminal velocity before it reaches the ground.

Describe how the forces acting on the sand-filled ball at v = 0.50 m s^–1 compare with the forces acting on the hollow ball at this speed.

Explain why the terminal velocity of the sand-filled ball will be greater than the terminal velocity of the hollow ball.

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.

Fig. 4.1 shows an arrangement used by a student to determine the acceleration of free fall.

A trolley is attached to a variable mass m by a string which passes over a pulley.

The mass m is released from rest and falls through a fixed height of 0.600 m accelerating the trolley of mass 0.800 kg. When the mass m hits the floor, the trolley then continues to move at a constant velocity v.

This constant velocity v is determined by measuring the time t for the card of length 0.200 m to pass fully through a light gate connected to a timer.

Frictional forces on the trolley and the falling mass m are negligible.

Show that the relationship between v and m is

where g is the acceleration of free fall.

The student records the information from the experiment in a table. The column headings and just the last row for m = 0.600 kg from this table are shown below.

It is suspected that the card on the trolley did not pass at right angles through the light beam.

Discuss, without doing any calculations, the effect this may have on the experimental value for the acceleration of free fall g.

A swimming pool designer investigates the depth d below a water surface reached by a diver when diving from a height h above the water surface. The designer models the diver as a uniform wooden cylinder. The experimental arrangement is shown in Fig. 18.1.

Calculate the density of the wood.

density =........ kg m⁻³

Suggest why wood is an appropriate material to model the depth reached by a diver.

The cylinder is released from rest from a trapdoor. The base of the cylinder is at a height h = 0.30 m above the water surface. Calculate the speed of the cylinder just before the base hits the water. Ignore air resistance.

speed =........ m s⁻¹

Fig. 18.2 shows the cylinder fully submerged under the water surface before it has come to rest. The cylinder is moving vertically down.

Add arrows to Fig. 18.2 to show the three forces acting on the wooden cylinder. Label

the arrows.

Describe and explain how the resultant force on the wooden cylinder varies from the moment the cylinder is fully submerged until it reaches its deepest point.

The graph of Fig. 18.3 shows the depth d reached for different initial drop height h.

The designer is required to double the height of a diving board for an existing swimming pool. He suggests that the depth of the pool also needs to be doubled. Use Fig. 18.3 to explain whether you agree with this suggestion.