A 64.4-kg skateboarder starts out with a speed of 2.11 m/s. He does 81.6 J of work on himself by pushing with his feet against the ground. In addition, friction does -289 J of work on him. In both cases, the forces doing the work are non-conservative. The final speed of the skateboarder is 8.56 m/s. (a) Calculate the change (PEf - PE0) in the gravitational potential energy. (b) How much has the vertical height of the skater changed? Give the absolute value.
Oh, gravity and heights, huh? Time to bring a little humor into the mix!
(a) To calculate the change in gravitational potential energy, we need to find the difference between the initial and final potential energy. It's as if the skater is playing a game of "hide and seek" with gravity!
Now, we know that potential energy is directly related to height. So, at the start, the skater must've been saying to gravity, "Height, height, go away, come again another day!" But in the end, gravity finally caught up, and the skater said, "Fine, I'll change my potential energy!"
Let's use the equation for gravitational potential energy: PE = mgh.
Since the mass (m) and gravity (g) are constant, we can calculate the change in potential energy by simply finding the difference in height (h) between the initial and final state.
(b) But how much has the vertical height of the skater changed? Well, it's like a hidden treasure hunt! The change in height is the difference in the skater's position from the start to the end. It's like finding the hidden loot!
To find the absolute value of the change in height, you need to subtract the initial height (h0) from the final height (hf). And just like magic, the absolute value will show you how high the skater has risen or fallen!
Remember, my friend, physics equations may seem daunting, but they hide a little fun in there too! Happy calculating!
To calculate the change in gravitational potential energy (PEf - PE0), we need to know the vertical height change (Δh) of the skater. Assuming the only forces acting on the skater are gravity and friction, we can calculate the change in gravitational potential energy using the following steps:
(a) Calculate the change (PEf - PE0) in gravitational potential energy:
1. First, let's calculate the initial gravitational potential energy (PE0) using the formula: PE0 = m * g * h0, where m is the mass of the skateboarder, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h0 is the initial vertical height.
PE0 = (64.4 kg) * (9.8 m/s^2) * h0
2. Next, let's calculate the final gravitational potential energy (PEf) using the formula: PEf = m * g * hf, where hf is the final vertical height.
PEf = (64.4 kg) * (9.8 m/s^2) * hf
3. Finally, calculate the change (PEf - PE0) in gravitational potential energy:
Change in Gravitational Potential Energy (PEf - PE0) = PEf - PE0
(b) Calculate the change in vertical height (Δh) of the skater:
Since the vertical height change (Δh) is not given explicitly, we can use the conservation of mechanical energy principle to determine it.
1. The work done on the skater by himself by pushing with his feet (W1) is given as 81.6 J.
2. The work done on the skater by friction (W2) is given as -289 J.
By using the principle of conservation of mechanical energy, we can write:
ΔPE = ΔKE + W1 + W2
where ΔPE is the change in potential energy, ΔKE is the change in kinetic energy, W1 is the work done by the skater, and W2 is the work done by friction.
In this case, the change in kinetic energy (ΔKE) is given by:
ΔKE = KEf - KE0 = (1/2) * m * (vf^2 - v0^2)
where m is the mass of the skateboarder, vf is the final velocity, and v0 is the initial velocity.
Since the initial velocity (v0) is 2.11 m/s and the final velocity (vf) is 8.56 m/s, we can calculate the change in kinetic energy (ΔKE).
Now, using the equation ΔPE = ΔKE + W1 + W2, substitute the given values to find the vertical height change (Δh).
Once we know Δh, we can find the absolute value of the change in height (|Δh|).
Please provide the values of the final vertical height (hf) and the initial vertical height (h0) to proceed with the calculations.
To calculate the change in gravitational potential energy (ΔPE), we can use the formula:
ΔPE = m * g * Δh
Where:
m = mass of the skateboarder = 64.4 kg
g = acceleration due to gravity = 9.8 m/s^2
Δh = change in vertical height
(a) Calculate the change in gravitational potential energy (ΔPE)
We need to find the change in potential energy between the initial state (PE0) and the final state (PEf). The change in potential energy is given by:
ΔPE = PEf - PE0
Given that the skateboarder's final speed is 8.56 m/s, we can assume that the final height (hf) is the same as the initial height (hi). This is because the change in gravitational potential energy is independent of the path taken by the skateboarder and only depends on the difference in height.
Therefore, ΔPE = PEf - PE0 can be simplified to:
ΔPE = m * g * (hf - hi)
Since hf = hi, we can rewrite it as:
ΔPE = m * g * Δh
Substituting the values:
ΔPE = 64.4 kg * 9.8 m/s^2 * Δh
(b) Calculate the change in height (Δh)
To find the absolute value of the change in height, we can rearrange the equation:
Δh = ΔPE / (m * g)
Substituting the known values:
Δh = 81.6 J / (64.4 kg * 9.8 m/s^2)
Now, calculate the value of Δh using a calculator:
Δh = 81.6 J / (631.12 kg⋅m²/s²)
Finally, calculate the absolute value of Δh, as height cannot be negative:
| Δh | = 0.1291 meters
Therefore:
(a) The change in gravitational potential energy is ΔPE = 64.4 kg * 9.8 m/s^2 * Δh = 64.4 kg * 9.8 m/s^2 * 0.1291 m = 82.67 J
(b) The absolute value of the change in height is | Δh | = 0.1291 meters.
ΔKE=1/2mvf^2- 1/2mv0^2
a)Plug numbers in
1. ΔKE=1/2(64.4)(8.56)^2- 1/2(64.4)(2.11)^2= -2216.05 (because displacement, I think)
2. -2216.05J+(-289J)-80J
Answer =-2425.05J
b.)
1. W=ΔKE, W=ΔPE(PE=mgh), ΔPE=ΔKE
(64.4kg)(9.8m/s^2)h=2425.05J
631.12h=2425.05J
Answer h=3.842m
Hope this Help
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