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.
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(b) How much has the vertical height of the skater changed?
To calculate the change in gravitational potential energy (ÄPE) of the skateboarder, we need to use the equation:
ÄPE = m * g * h
where:
m = mass of the skateboarder (57.5 kg)
g = acceleration due to gravity (approximately 9.8 m/s²)
h = change in vertical height
However, we are not given any information directly related to the change in height (h). So, to determine the change in gravitational potential energy, we will use the information we have about the change in kinetic energy and work done.
The change in kinetic energy (ÄKE) can be calculated as:
ÄKE = KEf - KE0
where:
KEf = final kinetic energy (½ * m * v²)
KE0 = initial kinetic energy (½ * m * u²)
Given that the initial speed (u) is 1.80 m/s and the final speed (v) is 6.20 m/s, we can calculate:
KEf = ½ * 57.5 kg * (6.20 m/s)² = ½ * 57.5 kg * 38.44 m²/s² = 1113.35 J
KE0 = ½ * 57.5 kg * (1.80 m/s)² = ½ * 57.5 kg * 3.24 m²/s² = 111.15 J
Substituting these values into the equation for ÄKE:
ÄKE = 1113.35 J - 111.15 J = 1002.20 J
Now, to find the change in potential energy (ÄPE), we can use the conservation of mechanical energy principle:
ÄKE + ÄPE = ÄW
where ÄW is the work done on the system. In this case, the work done on the skateboarder is +80.0 J by pushing with his feet and -265 J by friction. Therefore:
ÄW = 80.0 J + (-265 J) = -185.0 J
Rearranging the equation, we have:
ÄPE = ÄW - ÄKE = -185.0 J - 1002.20 J = -1187.20 J
Since the gravitational potential energy is a scalar quantity, the change in gravitational potential energy (ÄPE) is -1187.20 J.
Now, let's move on to part (b) of the question.
To determine the change in vertical height, we can rearrange the equation for gravitational potential energy:
ÄPE = m * g * h
Rearranging it to solve for h:
h = ÄPE / (m * g)
Substituting the values we have:
h = (-1187.20 J) / (57.5 kg * 9.8 m/s²) = -2.06 m
The negative sign indicates a decrease in height. Therefore, the vertical height of the skater has decreased by approximately 2.06 meters.