A 64.2-kg skateboarder starts out with a speed of 2.21 m/s. He does 98.0 J of work on himself by pushing with his feet against the ground. In addition, friction does -276 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.
To calculate the change in gravitational potential energy (ΔPE), you can use the equation:
ΔPE = m * g * Δh
where m is the mass of the skateboarder (64.2 kg), g is the acceleration due to gravity (9.8 m/s²), and Δh is the change in vertical height.
(a) To calculate the change in gravitational potential energy (ΔPE), we need to determine the initial potential energy (PE0) and the final potential energy (PEf). Since only the forces doing work are non-conservative, the change in potential energy will be given by the work done on the skateboarder.
The work done by the skateboarder (98 J, positive) and the work done by friction (-276 J, negative) add up to give the net work done:
Net Work = Work by the skateboarder + Work by friction
Net Work = 98 J - 276 J
Net Work = -178 J
The change in potential energy is equal to the net work done:
ΔPE = -178 J
(b) To find the absolute value of the change in vertical height, we can rearrange the equation for ΔPE:
ΔPE = m * g * Δh
Solving for Δh:
Δh = ΔPE / (m * g)
Δh = (-178 J) / (64.2 kg * 9.8 m/s²)
Δh ≈ -0.28 m
Taking the absolute value, the change in vertical height is approximately 0.28 m.