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 in the skis, how high was the hill?
A) 2.5 m
B) 0.25 m
C) 10 m
D) 7.35 m
D) 7.35 m
Well, let me put on my thinking ski jacket and crunch some numbers for you.
We can start by using the principle of conservation of mechanical energy, which states that energy is neither created nor destroyed, it just transforms from one form to another. In this case, we're dealing with gravitational potential energy and kinetic energy.
Now, if there's no friction, we can assume that no energy is lost due to heating up the slopes. So, the initial kinetic energy of the skier will be equal to the final gravitational potential energy at the top of the hill.
The initial kinetic energy (KE) of the skier is given by the equation KE = (1/2)mv^2, where m is the mass of the skier (60 kg) and v is the initial velocity (12 m/s).
KE = (1/2)(60 kg)(12 m/s)^2
KE = 4320 J
Now, at the top of the hill, all of the skier's initial kinetic energy has been transformed into gravitational potential energy (PE). The formula for gravitational potential energy is PE = mgh, where m is the mass of the skier (60 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the hill.
So, we have PE = (60 kg)(9.8 m/s^2)(h)
Now, we can equate the initial kinetic energy (4320 J) to the final gravitational potential energy by setting up the equation:
4320 J = (60 kg)(9.8 m/s^2)(h)
Now, let's solve for h:
h = 4320 J / (60 kg * 9.8 m/s^2)
h ≈ 7.35 m
Therefore, the height of the hill is approximately 7.35 meters.
So the answer is D) 7.35 m. But don't worry, the skier didn't go over the hill and fall like my hopes and dreams. It was a successful journey to the top!
To determine the height of the hill, we can use the principle of conservation of energy. At the bottom of the hill, the skier has kinetic energy given by:
KE1 = 1/2 * m * v1^2
where m is the mass of the skier and v1 is the initial velocity. At the top of the hill, the skier has gravitational potential energy given by:
PE2 = m * g * h
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the height of the hill. At the top of the hill, the skier also has kinetic energy given by:
KE2 = 1/2 * m * v2^2
where v2 is the final velocity at the top of the hill.
According to the principle of conservation of energy, the initial kinetic energy plus the initial potential energy is equal to the final kinetic energy plus the final potential energy:
KE1 + PE1 = KE2 + PE2
Since there is no friction, the initial and final kinetic energy are the same:
1/2 * m * v1^2 + PE1 = 1/2 * m * v2^2 + PE2
We can rearrange the equation to solve for the height of the hill:
PE2 = 1/2 * m * (v1^2 - v2^2)
Substituting the given values:
PE2 = 1/2 * 60 kg * (12 m/s)^2 - (9.75 m/s)^2
PE2 = 1/2 * 60 kg * (144 m^2/s^2) - (95.0625 m^2/s^2)
PE2 = 4320 J - 5703.75 J
PE2 = -1383.75 J (negative sign indicates the potential energy is decreasing)
The height of the hill can be found by dividing the potential energy by the gravitational acceleration:
h = PE2 / (m * g)
h = -1383.75 J / (60 kg * 9.8 m/s^2)
h ≈ -1383.75 J / 588 N
h ≈ -2.352 m
Since the height of the hill cannot be negative, we can conclude that the height of the hill is approximately 2.35 m.
Therefore, the correct answer is not given among the options provided.
To solve this problem, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of an object remains constant if there are no external forces doing work on it. Mechanical energy consists of two components: kinetic energy (KE) and potential energy (PE).
At the bottom of the hill, the skier has only kinetic energy, which can be calculated using the formula:
KE = 0.5 * mass * velocity^2
Given that the mass of the skier is 60 kg and the initial velocity is 12 m/s, we can calculate the initial kinetic energy:
KE_initial = 0.5 * 60 kg * (12 m/s)^2 = 4320 J
At the top of the hill, the skier has some potential energy due to their height above the ground. The potential energy (PE) of an object is given by the formula:
PE = mass * gravity * height
where gravity is approximately 9.8 m/s². Let's assume the height of the hill is h, so the potential energy at the top of the hill is:
PE_top = 60 kg * 9.8 m/s² * h = 588 h J
Since there is no friction, the total mechanical energy remains constant. Therefore, we can equate the initial kinetic energy to the potential energy at the top of the hill:
4320 J = 588 h J
To solve for height (h), we divide both sides of the equation by 588:
h = 4320 J / 588 J ≈ 7.35 m
Therefore, the height of the hill is approximately 7.35 meters.
The correct answer choice is D) 7.35 m.