A 64.4-kg skateboarder starts out with a speed of 1.95 m/s. He does 119 J of work on himself by pushing with his feet against the ground. In addition, friction does -260 J of work on him. In both cases, the forces doing the work are non-conservative. The final speed of the skateboarder is 8.68 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 principle of conservation of energy. The total mechanical energy of the skateboarder, which includes the kinetic energy (KE) and potential energy (PE), is conserved in the absence of any external work or non-conservative forces.

(a) Firstly, let's calculate the change in kinetic energy (ΔKE):

ΔKE = KEf - KE0
ΔKE = (1/2) * m * (vf^2 - v0^2)
= (1/2) * 64.4 kg * (8.68 m/s)^2 - (1.95 m/s)^2
= (1/2) * 64.4 kg * 63.57 m^2/s^2
= 2049.9368 J

(b) The work done by pushing with his feet is positive, implying that energy was added to the system. However, the work done by friction is negative, meaning energy was taken from the system. Therefore, the change in total mechanical energy (ΔE) is the sum of the work done by the skateboarder (Wp) and the work done by friction (Wf):

ΔE = Wp + Wf = 119 J + (-260 J) = -141 J

Since the total mechanical energy is conserved, the change in kinetic energy is equal to the change in total mechanical energy:

ΔKE = ΔE
2049.9368 J = -141 J

However, this is contradictory. According to the information provided, the change in kinetic energy does not match the change in total mechanical energy. There might be some missing or incorrect information in the problem statement. Please double-check the given parameters and reanalyze the problem to ensure accuracy.