Final answer:
To calculate the minimum speed v that you must give the box at the bottom of the incline so that it will reach the skier, you can use the work-energy theorem. The correct answer is a) v = sqrt(2gh/sinα).
Step-by-step explanation:
To calculate the minimum speed v that you must give the box at the bottom of the incline so that it will reach the skier, we can use the work-energy theorem. 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 equal to the change in its kinetic energy as it moves up the incline. The work done by the gravitational force is given by the formula W = mgh(sinα + μ_k cosα), where m is the mass of the box, g is the acceleration due to gravity, h is the height of the incline, μ_k is the coefficient of kinetic friction, and α is the angle of the incline.
Since the box starts from rest at the bottom of the incline, its initial kinetic energy is zero. Therefore, the work done by the gravitational force is equal to the final kinetic energy of the box:
W = mgh(sinα + μ_k cosα) = (1/2)mv²
Simplifying the equation, we can solve for v:
v = sqrt(2gh(sinα + μ_k cosα)/m)
Expressing the answer in terms of the given variables, we have v = sqrt(2gh(sinα + μ_k cosα)/m). Therefore, option a) v = sqrt(2gh/sinα) is the correct answer.