Gravitational Potential Energy & Kinetic Energy (CIE AS Physics)

Pumped hydroelectric systems store large bodies of water behind a dam. When electricity is needed in the grid, the dam is lifted, allowing water to flow from the upper reservoir.

One such power station, known as the Cruachan power station, is shown in Fig. 1.1. This facility has been producing hydroelectric power since 1965.

The vertical distance between the base of the upper reservoir and the turbine is 396 m. This arrangement allows Cruachan power station to produce a power output of 0.62 GW with an efficiency of 76%.

Determine the energy transferred by the water each second in order to achieve this power output.

Water is made to flow out of the upper reservoir through connecting pipes at a rate of 200 m³ per second. The density of water is 1.0 × 10³ kg m⁻³. 

Estimate the depth of the water held in the upper reservoir. State any assumptions made.

A BASE jumper of mass 70 kg jumps from the top of a bridge and reaches a speed of 45 m s⁻¹ after falling a distance of 150 m.

Calculate the change in the jumper's gravitational potential energy.

Calculate the kinetic energy gained by the jumper.

Explain why the values in (a) and (b) are not equal.

Calculate the average resistive force acting on the BASE jumper as they fall.

A car on a roller coaster has a mass of 4200 kg and is designed to carry up to 10 passengers each with an average mass of 65 kg.

The car is moving at a speed of 2.0 m s⁻¹ when it starts to descend through a drop of 50 m. It reaches a top speed of 24 m s⁻¹ after travelling 75 m along the track.

For the car with its maximum capacity of passengers, calculate

Calculate the kinetic energy of the car and passengers after the descent.

For the roller coaster car and passengers

Calculate the average resistive force that acts on the car during the descent.

Anaisha launches a pebble vertically upwards using a sling shot. The speed of the pebble as it leaves the sling shot is 28 m s⁻¹.

Calculate the height that the pebble gains.

Assume that air resistance is negligible. 

The mass of the pebble is 30 g. Complete the energy bar chart shown in Fig. 1.1. 

Fig 1.1

Anaisha notices a small basket hanging by a string from a tree. The basket has a mass of 28 g and the length of the string is 85 cm. Anaisha uses her slingshot to fire the pebble horizontally, which lands in the basket. The basket then swings out to an angle of 45°.

Calculate the speed of the pebble as she fired it.

Assume that air resistance is negligible and that there is no breeze.

Fig. 1.1 shows an object at rest at the top of a straight slope which makes a fixed angle with the horizontal at a distance h above the ground. 

Fig. 1.1

The object is released and slides down the slope from A to B with negligible friction.

Assume that the potential energy is zero at B. 

Sketch a graph showing: 

• The variation of potential energy along the slope. Label this P.
• The variation of kinetic energy of the object along the slope. Label this K.
• The variation of kinetic energy along the slope when there is a constant frictional force between the object and the surface. Label this F. 

Explain the important features of the graphs you have drawn.

In a theme park ride, a cage containing passengers falls freely a distance of  m from A to B and travels in a circular arc of radius 25 m from B to C. 

Brakes are applied at C after which the cage with its passengers travels 70 m along an upward sloping ramp and comes to rest at D. The track, together with relevant distances, is shown in Fig. 1.2. CD makes an angle of θ with the horizontal. 

Fig. 1.2

alt: For a passenger of mass 70 kg, the force keeping them in circular motion at C is 5.4 kN. 

The force required for circular motion on a passenger of mass 70 kg at C is 5.4 kN. 

Calculate height . 

Assume that friction is negligible between A and C.

The total mass of the cage and passengers is 720 kg. The average resistive force exerted by the brakes between C and D is 4.8 kN.

By considering energy changes, calculate the value of θ.