9)A skier starts from rest from the top of a 40 m high slope which makes 40 degrees with the ground. Coefficient of friction is 0.1 What is the velocity of the skier at the bottom of the ramp?

The velocity of the skier at the bottom of the ramp is approximately 26.288 meters per second.

Explanation:

We can determine the final velocity of the skier at the bottom of the ramp by Principle of Energy Conservation and Work-Energy Theorem, whose model is:

(1)

Where:

, - Initial and final gravitational potential energy, measured in joules.

, - Initial and final translational kinetic energy, measured in joules.

- Work dissipated by friction, measured in joules.

By definitions of gravitational potential and translational kinetic energy and work, we expand and simplify the model:

(2)

Where:

- Mass, measured in kilograms.

- Gravitational acceleration, measured in meters per square second.

, - Initial and final heights of the skier, measured in meters.

- Normal force from the incline on the skier, measured in newtons.

- Distance covered by the skier, measured in meters.

- Kinetic coefficient of friction, dimensionless.

The normal force exerted on the skier and the covered distance are, respectively:

(3)

(4)

Where is the angle of the incline above the horizontal, measured in sexagesimal degrees.

By applying (3) and (4) in (2), we get that:

(5)

Then, we clear the velocity of the skier at the bottom of the ramp is: (, , , , )

(6)

The velocity of the skier at the bottom of the ramp is approximately 26.288 meters per second.