A 60.8-kg skateboarder starts out with a speed of 1.84 m/s. He does 90.8 J of work on himself by pushing with his feet against the ground. In addition, friction does -278 J of work on him. In both cases, the forces doing the work are non-conservative. The final speed of the skateboarder is 8.12 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.
To solve this problem, we need to use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
(a) The change in gravitational potential energy (PE) can be calculated using the formula:
ΔPE = PEf - PE0
Given that the mass of the skateboarder is 60.8 kg and the acceleration due to gravity is 9.8 m/s^2, we can calculate the initial potential energy (PE0) using the formula:
PE0 = m * g * h
Where:
m = mass of the skateboarder = 60.8 kg
g = acceleration due to gravity = 9.8 m/s^2
h = initial vertical height
To calculate the final potential energy (PEf), we can use the formula:
PEf = m * g * H
Where:
H = final vertical height (the height is unknown and needs to be determined)
Since there is no initial or final vertical velocity given, it implies that the skater has moved vertically by some height H. Therefore, the initial and final kinetic energies would be zero.
Using the work-energy principle:
Work done by the skater on himself + Work done by friction = ΔKE
Given that the skater does 90.8 J of work on himself and friction does -278 J of work:
90.8 J + (-278 J) = 1/2 * m * (vf^2 - v0^2)
Where:
vf = final speed = 8.12 m/s
v0 = initial speed = 1.84 m/s
Let's solve the above equation to find the final kinetic energy (KEf):
90.8 J - 278 J = 1/2 * 60.8 kg * (8.12 m/s)^2 - 1/2 * 60.8 kg * (1.84 m/s)^2
12 = 1/2 * 60.8 kg * (8.12 m/s)^2 - 1/2 * 60.8 kg * (1.84 m/s)^2
12 = 1/2 * 60.8 kg * (66.0144 - 3.3856)
12 = 1/2 * 60.8 kg * 62.6288
12 = 1903.6288 kg⋅m^2/s^2
Thus, the final kinetic energy (KEf) is 1903.6288 kg⋅m^2/s^2.
As the initial and final potential energies are zero, we have:
ΔPE = PEf - PE0
ΔPE = 1903.6288 kg⋅m^2/s^2 - 0 kg⋅m^2/s^2
ΔPE = 1903.6288 kg⋅m^2/s^2
Therefore, the change in gravitational potential energy (PEf - PE0) is 1903.6288 kg⋅m^2/s^2.
(b) To find the change in vertical height, we can use the formula:
ΔPE = m * g * Δh
Where:
m = mass of the skateboarder = 60.8 kg
g = acceleration due to gravity = 9.8 m/s^2
Δh = change in vertical height (the height is unknown and needs to be determined)
Solving the equation for Δh:
1903.6288 kg⋅m^2/s^2 = 60.8 kg * 9.8 m/s^2 * Δh
Δh = 1903.6288 kg⋅m^2/s^2 / (60.8 kg * 9.8 m/s^2)
Δh = 3.0824 m
Therefore, the absolute value of the change in vertical height is 3.0824 meters.
To calculate the change in gravitational potential energy (PEf - PE0), we need to know the initial height of the skateboarder (PE0) and the final height (PEf). However, the given information does not provide any data regarding the heights involved. Therefore, we cannot calculate the change in gravitational potential energy without further information.
Similarly, determining the vertical height change also requires the initial and final heights. Without this information, it is not possible to determine the absolute value of the change in vertical height.