a 60 kg skier with an initial velocity of 12 m/s coasts up a hill. at the top of the hill, the skier is traveling at 9.75 m/s. Assuming there is no friction force in the skis, how high was the hill
A) 7.35 m
B) 0.25 m
C) 10 m
D) 2.5 m
GPE = KE (Gravitational Potential Energy = Kinetic Energy)
GPE = mgh (Gravitational Potential Energy = mass x gravity x height)
KE = 1/2mv^2
KE = 1/2(60) * (12)^2
KE = 4320
KE before hill: 4320 J
KE = 1/2(60) * (9.75)^2
KE = 2851.875
KE after hill: 2851.875 J
4320 - 2851.875 = 1468.125
1468.125 = mgh
1468.125 = (60)(9.81)(h)
1468.125/(60)(9.81) = h
1468.125/588.6 = h
2.494 = h
The answer is D. 2.5 m
Well, it seems like this skier is on quite the adventure! Let's see if we can calculate the height of the hill.
To solve this problem, we can use the principle of conservation of mechanical energy. As there is no friction force involved, the only forces at play are gravity and the initial velocity of the skier.
Now, we know that the skier's initial velocity is 12 m/s and their final velocity is 9.75 m/s. So there must have been a decrease in kinetic energy.
The change in kinetic energy can be calculated as:
ΔKE = KE final - KE initial
Where KE is the kinetic energy.
Now, we know that the change in kinetic energy is equal to the change in gravitational potential energy. The formula for gravitational potential energy is:
PE = m * g * h
Where m is the mass of the skier, g is the acceleration due to gravity, and h is the height of the hill.
Since there is no change in the skier's mass or the acceleration due to gravity, we can equate the change in kinetic energy to the change in potential energy:
ΔKE = m * g * h
Substituting the values we have:
(1/2) * m * (vf^2 - vi^2) = m * g * h
Now we can rearrange the equation to solve for h:
h = (1/2) * (vf^2 - vi^2) / g
Plugging in the given values:
h = (1/2) * (9.75^2 - 12^2) / 9.8
After doing the math, we find that h ≈ 7.35 m
So, the correct answer is A) 7.35 m.
Looks like that skier had quite the climb! Hope they enjoyed the view from up there!
To determine the height of the hill, we can use the principle of conservation of energy. At the bottom of the hill, the total mechanical energy (kinetic energy + potential energy) of the skier is:
E_total = KE_bottom + PE_bottom
Since the skier is at rest at the bottom of the hill, the kinetic energy is zero:
E_total = 0 + PE_bottom
At the top of the hill, the skier has both kinetic energy and potential energy:
E_total = KE_top + PE_top
Using the equation for kinetic energy (KE = 0.5 * mass * velocity^2) and potential energy (PE = mass * gravity * height), we can rewrite these equations:
0 + mass * gravity * height_bottom = 0.5 * mass * velocity_top^2 + mass * gravity * height_top
Since the mass is cancels out:
gravity * height_bottom = 0.5 * velocity_top^2 + gravity * height_top
We know the values for the mass (60 kg), velocity at the bottom (12 m/s), and velocity at the top (9.75 m/s). We also know that the acceleration due to gravity is approximately 9.8 m/s^2. To find the height difference between the bottom and top of the hill, we need to rearrange the equation:
height_top - height_bottom = (gravity * height_bottom - 0.5 * velocity_top^2) / gravity
Plugging in the values:
height_top - height_bottom = (9.8 * height_bottom - 0.5 * 9.75^2) / 9.8
height_top - height_bottom = (9.8 * height_bottom - 47.59) / 9.8
Simplifying the equation:
height_top - height_bottom = (9.8 * height_bottom) / 9.8 - (47.59 / 9.8)
height_top - height_bottom = height_bottom - 4.86
The height difference is equal to 4.86 meters. To find the height of the hill, we need to add the difference to the height at the bottom:
height_top = height_bottom + 4.86
Now we can calculate the height of the hill by substituting the value of the height at the bottom:
height_top = height_bottom + 4.86
height_top = 0 + 4.86
height_top = 4.86 meters
Therefore, the height of the hill is approximately 4.86 meters.
None of the given answer choices (A, B, C, D) matches the calculated height of the hill, so the correct answer is not provided in the given options.
m V^2 - m v^2 = 2 m g h
12^2 - 9.75^2 = 2 * 9.8 * h