You are designing a ski jump ramp for the next Winter Olympics. You need to calculate the vertical height hh from the starting gate to the bottom of the ramp. The skiers push off hard with their ski poles at the start, just above the starting gate, so they typically have a speed of 2.0 m/sm/s as they reach the gate. For safety, the skiers should have a speed of no more than 30.0 m/sm/s when they reach the bottom of the ramp. You determine that for a 88.0 kgkg skier with good form, friction and air resistance will do total work of magnitude 4000 JJ on him during his run down the slope.

50.35 m

Explanation:

Here is the complete question

You are designing a ski jump ramp for the next Winter Olympics. You need to calculate the vertical height (h) from the starting gate to the bottom of the ramp. The skiers push off hard with their ski poles at the start, just above the starting gate, so they typically have a speed of 2.0 m/s as they reach the gate. For safety, the skiers should have a speed of no more than 30.0 m/s when they reach the bottom of the ramp. You determine that for a 85.0 kg skier with good form, friction and air resistance will do total work of magnitude 4000 J on him during his run down the slope.

What is the maximum height (h) for which the maximum safe speed will not be exceeded?

Solution

From work-kinetic energy principles,

work done by gravity on skier, W₁ + work done by friction and air resistance on skier W₂ = kinetic energy change of skier ΔK.

W₁ + W₂ = ΔK

work done by gravity on skier,W₁ = mgh where m = mass of skier = 88.0 kg , g = 9.8 m/s² and h = vertical height of ramp

W₁ = 88 × 9.8h = 862.4h

work done by friction and air resistance on skier, W₂ = -4000 J

kinetic energy change of skier ΔK =¹/₂m(v₂² - v₁²) where v₁ = speed of skier at top of ramp = 2.0 m/s and v₂ = speed of skier at top of ramp = 30.0 m/s

ΔK =¹/₂ × 88(30² - 2²) = 44 × 896 = 39424 J

W₁ + W₂ = ΔK

862.4h + (-4000 J) = 39424 J

862.4h - 4000 J = 39424 J

862.4h = 39424 J + 4000 J

862.4h = 43424 J

h = 43424/862.4 = 50.35 m

So, the maximum height (h) for which the maximum safe speed will not be exceeded is 50.35 m

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