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 will first calculate the change in kinetic energy, then use that information to find the change in potential energy. Finally, we can determine the change in vertical height.
(a) Change in Kinetic Energy (ΔKE):
The change in kinetic energy is equal to the work done on the skateboarder. Therefore, we can calculate it by adding the work done by the skateboarder on himself and the work done by friction:
ΔKE = Work by Skateboarder + Work by Friction
ΔKE = 119 J + (-260 J) (Since work by friction is negative)
ΔKE = -141 J
The negative sign indicates that the change in kinetic energy is a decrease.
Change in Potential Energy (ΔPE):
The change in potential energy is equal to the negative change in kinetic energy (according to the law of conservation of energy):
ΔPE = - ΔKE
ΔPE = - (-141 J) (Negative sign cancels out)
ΔPE = 141 J
Therefore, the change in potential energy is 141 J.
(b) Change in Vertical Height (Δh):
The change in vertical height can be calculated using the change in potential energy. The formula for potential energy is:
PE = m * g * h (where m = mass, g = acceleration due to gravity, and h = height)
We want to find the change in height:
ΔPE = m * g * Δh
Δh = ΔPE / (m * g) (Solving for Δh)
Substituting the given values:
Δh = 141 J / (64.4 kg * 9.8 m/s²)
Δh ≈ 0.218 m
Therefore, the absolute value of the change in vertical height is approximately 0.218 m.