A skier starts from rest at point A and slides down the hill without turning or breaking. The friction coefficient is μ. When he stops at point B, his horizontal displacement is s. What is the height difference between points A and B? (The$$ velocity of the skier is small, so that the additional pressure on the snow due to the curvaturecan be neglected. Neglect also the friction of air and the dependence of μ on the velocity of the skier.$$)

Question

A skier starts from rest at point A and slides down the hill without turning or breaking. The friction coefficient is μ. When he stops at point B, his horizontal displacement is s. What is the height difference between points A and B? (The$$ velocity of the skier is small, so that the additional pressure on the snow due to the curvaturecan be neglected. Neglect also the friction of air and the dependence of μ on the velocity of the skier.$$)

Infinity Learn · Accepted Answer

According to question, the given situation is shown in the figure below.For a sufficiently $$safe-horizontal$$ displacement,  ∆s can be considered straight. If the corresponding length of path element is ∆L, the work done against frictional force is given byw∆$$F=$$μ$$mg($$ΔLcosθ$$)From$$   ΔOPQ,$$cos$$θ$$=$$ΔsΔLSo,   W∆$$F=$$μmgΔLΔsΔ$$L=$$μmgΔsOn adding up, we find that along the whole path, the total work done by the friction force is μmgs. By law of conservation of energy, this must equals to the decrease in potential energy of skier.∴  μ$$mgs=mghHence$$,    $$h=$$μs

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