A 55 kg skier is at the top of a slope that is 10.0 m high.
a) Set the zero level for gravitational potential energy at the bottom of the slope and find the gravitational potential energy for the skier at the top and the bottom of the slope. What is the PE difference between these two points?
b) Now repeat this calculation with the zero level at the top of the slope.
c) Repeat this problem with the zero level midway down the slope, at a height of 5.0 m.
a) To find the gravitational potential energy at the top and bottom of the slope, we can use the formula:
Gravitational potential energy (PE) = mass × gravity × height
Given:
Mass of the skier (m) = 55 kg
Height of the slope (h) = 10.0 m
Acceleration due to gravity (g) = 9.8 m/s^2
At the top of the slope, the PE is given by:
PE_top = m × g × h
= 55 kg × 9.8 m/s^2 × 10.0 m
= 5390 J
At the bottom of the slope, the PE is zero since we set the zero level for gravitational potential energy there.
The PE difference between the top and bottom points is:
PE_difference = PE_top - 0
= 5390 J - 0
= 5390 J
Therefore, the PE difference between these two points is 5390 Joules.
b) To calculate the gravitational potential energy with the zero level at the top of the slope, the PE at the top and bottom points will be different.
At the top of the slope, the PE is zero since we set the zero level there.
PE_top = 0 J
At the bottom of the slope, the PE is given by:
PE_bottom = m × g × h
= 55 kg × 9.8 m/s^2 × 10.0 m
= 5390 J
The PE difference between the top and bottom points is:
PE_difference = PE_bottom - PE_top
= 5390 J - 0 J
= 5390 J
Therefore, the PE difference between these two points is again 5390 Joules.
c) To calculate the gravitational potential energy with the zero level midway down the slope, at a height of 5.0 m, the PE at the top and bottom points will be different.
At the top of the slope (relative to the new zero level), the height is 10.0 m - 5.0 m = 5.0 m.
PE_top = m × g × h
= 55 kg × 9.8 m/s^2 × 5.0 m
= 2695 J
At the bottom of the slope (relative to the new zero level), the height is 0 m - 5.0 m = -5.0 m.
PE_bottom = m × g × h
= 55 kg × 9.8 m/s^2 × -5.0 m
= -2695 J
The PE difference between the top and bottom points is:
PE_difference = PE_bottom - PE_top
= -2695 J - 2695 J
= -5390 J
Therefore, the PE difference between these two points is -5390 Joules.
a) Well, well, well, we have a skier and a slope, a match made in snow heaven! Let's get calculating. So, the skier is 10.0 m high at the top of the slope. Since the zero level for gravitational potential energy is at the bottom, we can say that the skier's potential energy at the top is... well, zero! Gravity hasn't had a chance to show off yet.
Now let's roll down the slope and see what happens. As the skier reaches the bottom, the potential energy reaches its maximum value. We can calculate this using the formula PE = mgh, where m is the mass (55 kg), g is the acceleration due to gravity (don't worry, it's always -9.8 m/s²), and h is the height (10.0 m). Crunching those numbers, we find that the skier's potential energy at the bottom is 5,445 J.
Now, to find the potential energy difference between these two points, we simply subtract the potential energy at the top from the potential energy at the bottom. So, 5,445 J - 0 J = 5,445 J. Voila! That's the PE difference.
b) Alright, now we're going topsy-turvy with the zero level at the top. The skier's potential energy at the top is still zero, because we said so. So the real question is, what's the potential energy at the bottom?
Using the same formula as before, PE = mgh, we can plug in the numbers and calculate. This time, h is still 10.0 m, but g gets a special treatment. Remember, since the zero level is at the top, the acceleration due to gravity is now 9.8 m/s², positive and proud. Crunching those numbers once again, we find that the skier's potential energy at the bottom is -5,445 J.
Don't get confused by the negative sign. It just means that the potential energy is now negative because we've reversed our gravitational orientation. To find the PE difference, we can subtract the potential energy at the top (which is still zero) from the potential energy at the bottom (-5,445 J). That gives us a final answer of -5,445 J. Negative potential energy... downhill, indeed!
c) Now we're dropping the zero level smack in the middle of the slope, at a height of 5.0 m. The top of the slope is still our reference point for a potential energy of zero.
To find the potential energy at this new zero level, we'll have to calculate the difference in height between our new zero level (5.0 m) and the top (10.0 m). That gives us a difference in height of -5.0 m (because we're going downhill here). Again, don't be fooled by the negative sign, it's just showing that the skier is lower than our reference point.
Using the formula PE = mgh, we can find the potential energy at this midway point. Plugging in the numbers, m = 55 kg, g = -9.8 m/s² (negative because we're still going downhill), and h = -5.0 m. Crunching those numbers, we find that the skier's potential energy at the midpoint is -2,722.5 J.
And now for the PE difference between this halfway point and the bottom. We know that the potential energy at the bottom is still -5,445 J (because we calculated it in part b). So, to find the PE difference, we subtract the potential energy at the midway point (-2,722.5 J) from the potential energy at the bottom (-5,445 J). This gives us a final answer of -2,722.5 J. And with that, we've mastered the art of potential energy on a clown's favorite ski slope! Enjoy the ride!
a) To calculate the gravitational potential energy of the skier at the top and bottom of the slope, we can use the formula:
Gravitational Potential Energy = mass * acceleration due to gravity * height
First, let's find the gravitational potential energy at the top of the slope. We are given that the skier has a mass of 55 kg and the height of the slope is 10.0 m. The acceleration due to gravity is approximately 9.8 m/s^2.
Gravitational Potential Energy at the top = 55 kg * 9.8 m/s^2 * 10.0 m = 5390 J
Next, let's find the gravitational potential energy at the bottom of the slope. Since the zero level for gravitational potential energy is set at the bottom, the height at the bottom is 0.
Gravitational Potential Energy at the bottom = 55 kg * 9.8 m/s^2 * 0 m = 0 J
To find the potential energy difference between the top and bottom, we subtract the potential energy at the bottom from the potential energy at the top:
PE difference = 5390 J - 0 J = 5390 J
Therefore, the potential energy difference between the top and bottom of the slope is 5390 Joules.
b) Now let's calculate the gravitational potential energy with the zero level at the top of the slope.
Using the same formula:
Gravitational Potential Energy = mass * acceleration due to gravity * height
First, we find the potential energy at the top of the slope. Since the zero level is at the top, the height at the top is 0.
Gravitational Potential Energy at the top = 55 kg * 9.8 m/s^2 * 0 m = 0 J
Next, let's find the potential energy at the bottom. The height at the bottom is 10.0 m.
Gravitational Potential Energy at the bottom = 55 kg * 9.8 m/s^2 * 10.0 m = 5390 J
To find the potential energy difference, we subtract the potential energy at the top from the potential energy at the bottom:
PE difference = 5390 J - 0 J = 5390 J
Therefore, the potential energy difference between the top and bottom of the slope is 5390 Joules, regardless of the choice of the zero level.
c) Now let's calculate the gravitational potential energy with the zero level midway down the slope, at a height of 5.0 m.
First, let's find the potential energy at the top of the slope. The height at the top is 10.0 m - 5.0 m = 5.0 m.
Gravitational Potential Energy at the top = 55 kg * 9.8 m/s^2 * 5.0 m = 2695 J
Next, let's find the potential energy at the bottom of the slope. The height at the bottom is 0 - 5.0 m = -5.0 m. Note that negative height means below the chosen zero level.
Gravitational Potential Energy at the bottom = 55 kg * 9.8 m/s^2 * (-5.0 m) = -2695 J
To find the potential energy difference, we subtract the potential energy at the bottom from the potential energy at the top:
PE difference = 2695 J - (-2695 J) = 2695 J + 2695 J = 5390 J
Therefore, the potential energy difference between the top and bottom of the slope is again 5390 Joules, regardless of the choice of the zero level.