A ski lift has a one-way length of 1 km and a vertical rise of 200 m. The chairs are spaced 20 m apart, and each chair can seat three people. The lift is operating at a steady speed of 10 km/h. Neglecting friction and air drag, and assuming that the average mass of each loaded chair is 250 kg, determine the power required to operate this ski lift. Also, estimate the power required to accelerate this ski lift in 17 s to its operating speed when it is first turned on.

Explanation:

The question states that the chairs are spaced 20 m apart through a length of 1 km, or say, 1000 m.

It also does say that each chair weighs 250 kg, and as such the load is

M = 50 * 250

M = 12500.

Taking into consideration, the initial and final heights, we have

h1 = 0, h2 = 200 m

The work needed to raise the chairs,

W = mgh, where h = h2 - h1

W = 12500 * 9.81 * (200 - 0)

W = 2.54*10^7 J

The work is done at a rate of 10 km/h, and at a distance of 1 km, time taken would be

t = 1/10 = 0.1 h or say, 360 s

The power needed thus, is

P = W/t

P = 2.54*10^7 / 360

P = 68125 W, or 68 kW

Initial velocity, u = 0 m/s

Final velocity, v = 10 km/h = 2.78 m/s

Startup time, t is 17 s

Acceleration during the startup then is

a = (v - u)/t

a = 2.78/17

a = 0.163 m/s²

The power needed for the acceleration is

P = ½m [(v² - u²)/t]

P = ½ * 12500 * [2.78²/17]

P = 6250 * 0.455

P = 2844 W

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