A 75.0-kg cross-country skier is climbing a 3.0o
slope at a constant speed of 2.00 m/s and
encounters air resistance of 25.0 N. Find his power output for work done against the gravitational
force and air resistance. (b) What average force does he exert backward on the snow to accomplish
this? (c) If he continues to exert this force and to experience the same air resistance when he
reaches a level area, how long will it take him to reach a velocity of 10.0 m/s?
To find the power output of the skier, we need to calculate the work done against the gravitational force and air resistance.
(a) Power output can be calculated using the formula:
Power = Force * Velocity
The work done against gravitational force can be calculated using the formula:
Work = Force * Distance
Since the skier is climbing a slope, the gravitational force can be calculated using:
Force of gravity = mass * gravity * sin(theta)
where:
mass = 75.0 kg (mass of the skier)
gravity = 9.8 m/s^2 (acceleration due to gravity)
theta = 3.0 degrees (slope angle)
Therefore, the force of gravity is:
Force of gravity = 75.0 kg * 9.8 m/s^2 * sin(3.0o)
Next, let's calculate the work done against the gravitational force and air resistance:
Work = (Force of gravity + Air resistance) * Distance
Since the skier is moving at a constant velocity, the work done against air resistance is:
Work against air resistance = Air resistance * Distance
Given the speed of the skier is 2.00 m/s, we can make use of the formula:
Distance = Speed * Time
To find the power output, we need to equate the work done against the gravitational force and air resistance with the given velocity:
Power = (Force of gravity + Air resistance) * Distance / Time
(b) The average force exerted by the skier can be determined using Newton's second law:
Force = mass * acceleration
Since the skier is moving with a constant velocity, the acceleration is zero. Therefore, the average force exerted backward on the snow is equal to the force of air resistance.
(c) To calculate the time it will take for the skier to reach a velocity of 10.0 m/s, we can rearrange the equation for distance:
Distance = Speed * Time
And then, substitute the values into the equation for Work:
Work = (Force of gravity + Air resistance) * Distance
and rearrange it to solve for time:
Time = Work / [(Force of gravity + Air resistance) * Speed]
Substituting the known values will give us the time required.
By following these steps and calculations, you can find the power output, average force exerted on the snow, and the time taken by the skier to reach a velocity of 10.0 m/s.