A 57.5-kg skateboarder starts out with a speed of 1.80 m/s. He does +80.0 J of work on himself by pushing with his feet against the ground. In addition, friction does -265 J of work on him. In both cases, the forces doing the work are nonconservative. The final speed of the skateboarder is 6.20 m/s.
(a) Calculate the change (ÄPE = PEf - PE0) in the gravitational potential energy.
J
(b) How much has the vertical height of the skater changed?
m
To calculate the change in gravitational potential energy (ΔPE), we can use the following formula:
ΔPE = m * g * Δh
where:
m = mass of the skateboarder (57.5 kg)
g = acceleration due to gravity (9.8 m/s², approximate value on Earth)
Δh = change in vertical height (unknown)
(a) Calculate the change in gravitational potential energy:
ΔPE = m * g * Δh
To find Δh, we need to rearrange the formula:
Δh = ΔPE / (m * g)
Substituting the given values:
Δh = (80 J + (-265 J)) / (57.5 kg * 9.8 m/s²)
Simplify the equation:
Δh = (-185 J) / (563.5 kg⋅m/s²)
Calculating Δh:
Δh = -0.328 meters
Therefore, the change in gravitational potential energy (ΔPE) is -0.328 joules.
(b) To determine the change in vertical height, we can use the formula:
ΔPE = m * g * Δh
Rearranging the formula:
Δh = ΔPE / (m * g)
Substituting the known values:
Δh = (80 J + (-265 J)) / (57.5 kg * 9.8 m/s²)
Simplifying the equation:
Δh = (-185 J) / (563.5 kg⋅m/s²)
Calculating Δh:
Δh = -0.328 meters
Therefore, the vertical height of the skater has changed by -0.328 meters.
To solve this problem, we need to calculate the change in gravitational potential energy and the change in vertical height of the skateboarder. Here's how:
(a) Calculate the change (ΔPE = PEf - PEi) in gravitational potential energy:
The gravitational potential energy can be calculated using the formula PE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
Given:
Initial speed (vi) = 1.80 m/s
Final speed (vf) = 6.20 m/s
Mass (m) = 57.5 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Using the equation for kinetic energy, we can find the initial and final kinetic energy:
Initial kinetic energy (KEi) = (1/2)mv^2 = (1/2)(57.5 kg)(1.80 m/s)^2
Final kinetic energy (KEf) = (1/2)mv^2 = (1/2)(57.5 kg)(6.20 m/s)^2
Now we can calculate the initial and final potential energies:
Initial potential energy (PEi) = mgh
Final potential energy (PEf) = mgh
To calculate the change in potential energy, we subtract the initial potential energy from the final potential energy:
ΔPE = PEf - PEi = mgh - mgh = 0
Therefore, the change in gravitational potential energy is zero (0 J).
(b) Calculate the change in vertical height (Δh) of the skater:
To find the change in vertical height, we can use the work-energy principle, which states that the work done on an object is equal to the change in energy of the object.
The work done on the skater by the external forces can be calculated by summing the work done by the skater on himself and the work done by friction:
External work (Wext) = Work done by the skater (Wskater) + Work done by friction (Wfriction)
Given:
Work done by the skater (Wskater) = 80.0 J
Work done by friction (Wfriction) = -265 J
Substituting these values, we get:
Wext = Wskater + Wfriction = 80.0 J - 265 J = -185 J
Since work W = ΔPE, we can equate the external work to the change in potential energy:
-185 J = ΔPE
Since we found that ΔPE = 0 J in part (a), the change in vertical height (Δh) is also zero (0 m).
Therefore, the change in vertical height of the skater is zero (0 m).