Final answer:
The change in the skateboarder's gravitational potential energy is 877.16 J, which corresponds to a vertical height change of approximately 1.67 m.
Step-by-step explanation:
The question asked is to calculate the change in the gravitational potential energy (PEF - PE0) of a skateboarder, and how much the vertical height of the skater has changed, based on the given work done on the skateboarder and the change in his speed.
Solution for (a)
To find the change in gravitational potential energy (APEg), we first need to compute the skateboarder's total change in kinetic energy (AKE), as the work done on him is non-conservative. We consider the work-energy principle, which states that the net work done on an object is equal to its change in kinetic energy (AKE = KF - K0).
The initial kinetic energy (K0) is (1/2)*m*v0^2 and the final kinetic energy (KF) is (1/2)*m*vf^2, where m is the mass of the skateboarder, v0 is the initial speed, and vf is the final speed.
Substituting the given values: K0 = 0.5*54.0 kg*(1.79 m/s)^2 and KF = 0.5*54.0 kg*(6.39 m/s)^2. Calculating these yield K0 = 86.79 J and KF = 1103.95 J. Therefore, AKE = KF - K0 = 1103.95 J - 86.79 J = 1017.16 J.
The net work done on the skateboarder is the work he did on himself plus the work done by friction: Net Work = Work by skater + Work by friction = 114 J - 254 J = -140 J. By the work-energy principle, this net work equals the change in kinetic energy.
Net Work = AKE, so -140 J = 1017.16 J - (PEF - PE0).
To find PEF - PE0, we rearrange the equation: (PEF - PE0) = AKE + Work by friction = 1017.16 J - 140 J = 877.16 J.
Solution for (b)
The change in gravitational potential energy is directly related to the change in height: APEg = mgh. So h = APEg / (mg).
Substituting the values: h = 877.16 J / (54.0 kg * 9.81 m/s^2) gives us h ≈ 1.67 m. So the vertical height of the skateboarder has changed by approximately 1.67 meters.