A 57.4-kg skateboarder starts out with a speed of 2.46 m/s. He does 114 J of work on himself by pushing with his feet against the ground. In addition, friction does -248 J of work on him. In both cases, the forces doing the work are non-conservative. The final speed of the skateboarder is 8.30 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 (PE) and vertical height (h) of the skateboarder, we can use the formulas:
(a) ΔPE = m * g * Δh
(b) Δh = ΔPE / (m * g)
Where:
ΔPE is the change in gravitational potential energy
m is the mass of the skateboarder (57.4 kg)
g is the acceleration due to gravity (approximately 9.8 m/s²)
Δh is the change in vertical height
Let's calculate each part:
(a) Change in gravitational potential energy (ΔPE):
First, we need to find the initial and final potential energy.
We know that the change in kinetic energy (ΔKE) is equal to the work done on the system:
ΔKE = Work non-conservative + Work gravity
Then, we can calculate the initial kinetic energy (KE0) and the final kinetic energy (KEf) using the formulas:
KE0 = (1/2) * m * v0^2
KEf = (1/2) * m * vf^2
Given:
Mass (m) = 57.4 kg
Initial Speed (v0) = 2.46 m/s
Final Speed (vf) = 8.30 m/s
KE0 = (1/2) * 57.4 kg * (2.46 m/s)^2
KEf = (1/2) * 57.4 kg * (8.30 m/s)^2
Next, we can calculate the change in gravitational potential energy (ΔPE):
ΔPE = KEf - KE0
(b) Change in vertical height (Δh):
Using the formula for Δh mentioned earlier:
Δh = ΔPE / (m * g)
Now let's calculate:
(a) Change in gravitational potential energy (ΔPE):
KE0 = (1/2) * 57.4 kg * (2.46 m/s)^2
KEf = (1/2) * 57.4 kg * (8.30 m/s)^2
Calculate ΔPE:
ΔPE = KEf - KE0
(b) Change in vertical height (Δh):
Δh = ΔPE / (m * g)
Substitute the values and calculate the change in vertical height (Δh).
Please note that the negative sign (-) in the work done by friction indicates work done on the system (against the motion).
I hope this helps! Let me know if you need any further assistance.