A skier travelling 37 mph encounters a 21 degree slope. With friction not considered, how far up the slope will he travel
iniital KE=final PE
1/2 m v^2=mgh/sin21
change mph to m/s before you start. Then, solve for h.
To find out how far the skier will travel up the slope, we can use the concept of work-energy theorem. According to this theorem, the work done on an object is equal to the change in its kinetic energy.
In this case, the work done on the skier will be equal to the change in his kinetic energy as he goes from his initial speed of 37 mph to a stop. The change in kinetic energy can be calculated as:
ΔKE = (1/2) * m * (vf^2 - vi^2)
Where ΔKE is the change in kinetic energy, m is the mass of the skier (which we don't have), vf is the final velocity (which is 0 since the skier comes to a stop), and vi is the initial velocity (37 mph).
However, since the mass of the skier is not provided, we cannot directly calculate the change in kinetic energy.
Alternatively, if we assume a specific mass for the skier (let's say 70 kg), we can convert the initial velocity to meters per second and calculate the change in kinetic energy. Then, using the work-energy theorem, we can determine the distance the skier travels up the slope.
First, let's convert the initial velocity from mph to m/s:
37 mph * (1609.34 m/1 mile) * (1 hour/3600 seconds) = 16.55 m/s (rounded to two decimal places)
Next, calculate the change in kinetic energy:
ΔKE = (1/2) * m * (vf^2 - vi^2)
ΔKE = (1/2) * 70 kg * (0 m/s)^2 - (16.55 m/s)^2
ΔKE = (1/2) * 70 kg * (-16.55 m/s)^2 (simplifying the equation)
Assuming there is no friction, all of the initial kinetic energy will be converted into potential energy as the skier reaches the top of the slope. This potential energy change will be equal to the work done on the skier, which can be calculated as:
Work = ΔPE = m * g * h
Where ΔPE is the change in potential energy, m is the mass of the skier (70 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height the skier travels up the slope (the value we are trying to find).
Now, equating work done to the change in potential energy:
ΔPE = Work
m * g * h = ΔKE
Substituting the values:
70 kg * 9.8 m/s^2 * h = (1/2) * 70 kg * (-16.55 m/s)^2
Now, we can solve for h:
h = [(1/2) * (-16.55 m/s)^2] / (9.8 m/s^2)
h ≈ 14.0 meters
Therefore, assuming a mass of 70 kg for the skier, they would travel approximately 14.0 meters up the slope. Keep in mind that this is a hypothetical scenario, as we do not have the actual mass of the skier to provide an accurate answer.