A skier traveling at 30.9 m/s encounters a 21.5 degree slope. If you could ignore friction, to the nearest meter, how far up the hill does he go?
h = (V^2-Vo^2)/2g
h = (o-(30.9)^2) / -19.6 = 49 m.
To determine how far the skier goes up the hill, we can use the concept of work and energy. The work done on the skier is equal to the change in his potential energy.
The formula for work is:
Work = Force × Distance × cos(θ),
where
- Work is the energy transferred to an object by a force acting on it,
- Force is the force applied on the object,
- Distance is the distance over which the force is applied,
- θ is the angle between the force and the displacement.
In this case, the force we want to consider is the component of the skier's weight parallel to the slope, which is given by:
Force = Weight × sin(θ),
where
- Weight is the force exerted on the skier due to gravity,
- θ is the angle of the slope.
To solve this problem, we can break down the problem into two components: the parallel component and the perpendicular component.
The parallel component of the force acting on the skier is:
Force_parallel = Weight × sin(θ).
The distance covered by the skier up the hill can be found by solving for distance in the work formula:
Distance = Work / (Force_parallel × cos(θ)).
First, let's find the weight of the skier using the formula:
Weight = mass × acceleration due to gravity.
Assuming the mass of the skier is 75 kg and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the weight:
Weight = 75 kg × 9.8 m/s².
Next, we can calculate the parallel force:
Force_parallel = Weight × sin(θ).
Given the angle θ of 21.5 degrees, we can calculate the parallel force:
Force_parallel = Weight × sin(21.5°).
Finally, we can calculate the distance using the work formula:
Distance = Work / (Force_parallel × cos(θ)).
Since we are ignoring friction in this scenario, the work done is equal to the change in potential energy.
Change in potential energy = W = m × g × h,
where
- m is the mass of the skier,
- g is the acceleration due to gravity,
- h is the vertical height (distance up the hill).
To solve for the distance up the hill, we can rearrange the equation:
h = W / (m × g).
Let's plug in the values and calculate:
h = (m × g × Distance) / (m × g × sin(θ) × cos(θ)).
Now, substitute the given values:
h = Distance / (sin(θ) × cos(θ)).
Given θ of 21.5 degrees, we can plug in the values and calculate the distance up the hill:
h = Distance / (sin(21.5°) × cos(21.5°)).
Solving this equation will give us the distance up the hill to the nearest meter.