You are a member of an Alpine Rescue Team. You must project a box of supplies up an incline ofconstant slope angle α so that it reaches a stranded skier who is a vertical distance h above the bottom ofthe incline. The incline is slippery, but there is some friction present, with kinetic friction coefficient µk .Use the work-energy theorem to calculate the minimum speed you must give the box at the bottom of theincline so that it will reach the skier. Express your results in terms of g, h, µ k and α.
Question
You are a member of an Alpine Rescue Team. You must project a box of supplies up an incline ofconstant slope angle α so that it reaches a stranded skier who is a vertical distance h above the bottom ofthe incline. The incline is slippery, but there is some friction present, with kinetic friction coefficient µk .Use the work-energy theorem to calculate the minimum speed you must give the box at the bottom of theincline so that it will reach the skier. Express your results in terms of g, h, µ k and α.
Solution
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done on the box is the sum of the work done by gravity and the work done by friction.
The work done by gravity is equal to the weight of the box times the vertical distance it travels, which is mgh, where m is the mass of the box, g is the acceleration due to gravity, and h is the vertical distance.
The work done by friction is equal to the force of friction times the distance the box travels along the incline. The force of friction is equal to the normal force (which is the weight of the box times the cosine of the slope angle) times the coefficient of kinetic friction. The distance the box travels along the incline is h/sin(α). Therefore, the work done by friction is µkmg*cos(α)*h/sin(α).
The change in kinetic energy of the box is equal to the final kinetic energy minus the initial kinetic energy. Since the box starts at rest, its initial kinetic energy is zero. Therefore, the change in kinetic energy is 1/2mv^2, where v is the final speed of the box.
Setting the work done equal to the change in kinetic energy and solving for v gives:
mgh + µkmgcos(α)h/sin(α) = 1/2mv^2
Solving for v gives:
v = sqrt[2gh(1 + µk*cos(α)/sin(α))]
This is the minimum speed you must give the box at the bottom of the incline so that it will reach the skier.
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