A mountain cable car carries 30 passengers up a slope. The total mass of the car and passengers is 4000 kg. The cable car travels at a constant speed of 5.0...

Marking Scheme

• (a) Distance = speed x time = \( 5.0 \times 180 = 900 \) m [1]
• (b) Height = \( 900 \times \sin 30° = 900 \times 0.50 = 450 \) m [1]
• (c) GPE = \( mgh = 4000 \times 10 \times 450 \) [1]; \( = 18\,000\,000 \) J (18 MJ) [1]
• (d) Force along slope = \( mg \sin 30° + \text{friction} = 4000 \times 10 \times 0.50 + 2000 \) [1]; \( = 22\,000 \) N [1]
• (e) \( P = Fv = 22\,000 \times 5.0 \) [1]; \( = 110\,000 \) W (110 kW) [1]
• (f) An emergency braking system / backup cable / automatic stop if cable tension drops [1]

Explanation

(a) At constant speed, distance = speed x time: \( d = 5.0 \times 180 = 900 \) m along the cable.

(b) The cable is inclined at 30° to the horizontal. The vertical height gained is the component of the 900 m distance in the vertical direction: \( h = 900 \times \sin 30° = 900 \times 0.50 = 450 \) m.

(c) The gravitational potential energy gained by the car and passengers is: \( \text{GPE} = mgh = 4000 \times 10 \times 450 = 18\,000\,000 \) J = 18 MJ. This is the minimum energy the motor must supply to raise the load (ignoring friction).

(d) The cable must exert enough force along the slope to overcome two opposing forces: the component of weight acting down the slope (\( mg \sin 30° = 4000 \times 10 \times 0.50 = 20\,000 \) N) and friction (2000 N). Since the car moves at constant speed, acceleration is zero, so the cable force equals the total opposing force: \( 20\,000 + 2000 = 22\,000 \) N.

(e) Power is force times velocity for constant-speed motion: \( P = Fv = 22\,000 \times 5.0 = 110\,000 \) W = 110 kW. This is the mechanical power the motor must deliver to the cable.

(f) Cable car systems carry passengers at great height, so safety is critical. An emergency braking system can grip the cable or track if the drive motor fails. Other reasonable features include a backup cable to prevent free fall, automatic shutdown if cable tension drops below a threshold, or a secondary power supply for the braking system.