Equations of uniformly accelerated motion

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.

A scientist uses a rocket to study the Earth’s atmosphere.

The scientist launches the rocket from rest at the point A at t = 0 seconds. The force produced by the rocket’s engine causes it to accelerate.

At t = 50 seconds, the rocket’s engine no longer produce an accelerating force as all of the fuel has been used. The rocket has reached point B. Its velocity is now 3.1 km s^–1 at an angle of 75° to the horizontal.

The rocket engine works by expelling hot gas backwards.

Explain, using Newton’s laws of motion, how the engine causes the rocket to accelerate between t = 0 and t = 50 s.

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

19* A student makes a pendulum using a length of string with a ball of adhesive putty which acts as a bob. The mass of this bob is M. A similar second pendulum is constructed with the same length of string but with a bob of a smaller mass. The mass of this bob is m.

The arrangement of the pendulums is shown below.

The bob of mass M is pulled back to a vertical height of H from its rest position. It is released and collides with the bob of mass m. The two bobs then stick together and reach a maximum vertical height h from the rest position.

Describe how to perform an experiment to test the validity of this equation and how the data can

be analysed.

Additional answer space if required

A car is travelling along a straight road at 18 m s^–1. 19 (a) The driver sees an obstacle and after 0.50 s applies the brakes. The stopping distance of the car is 38 m.

Calculate the magnitude of the deceleration of the car when the brakes are applied.

deceleration =........ m s^–2

A ball is dropped from rest above the ground. Air resistance has negligible effect on the motion of the ball.

The speed of the ball is v after it has fallen a distance h from its point of release.

Which graph is correct for this falling ball?

Your answer

An object above the ground is released from rest at time t = 0.

Air resistance is negligible.

What is the distance travelled by the object between t = 0.20 s and t = 0.30 s?

A 0.20 m

B 0.25 m

C 0.44 m

D 0.49 m

Your answer

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.