A 60.8-kg skateboarder starts out with a speed of 1.84 m/s. He does 90.8 J of work on himself by pushing with his feet against the ground. In addition, friction does -278 J of work on him. In both cases, the forces doing the work are non-conservative. The final speed of the skateboarder is 8.12 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.
He must have been heading downhill !
change in kinetic energy = (1/2)(60.8)(8.12^2 - 1.84^2 ) = 1902 Joules
energy in = -278 + 90.8 = -187.2
h = 0 initially
m g h = - 1902 + 187 = - 1715 Joules
h = -1715/(60.8*9.81)
= - 2.88 meters so 2.88 is absolute value
sorry, had to drop further to make up for friction
m g h = -1902 -187 = -2089 Joules
and h = 2089/(60.8*9.81)
To calculate the change in gravitational potential energy (PE) in part (a), we can use the formula:
ΔPE = PEf - PE0
where PEf is the final potential energy and PE0 is the initial potential energy.
The change in potential energy can be calculated using the formula:
ΔPE = m * g * h
where m is the mass of the skateboarder, g is the acceleration due to gravity, and h is the change in vertical height.
(a) Calculate the change in gravitational potential energy (PE):
To find the change in potential energy, we need to know the height difference between the initial and final positions. However, the given information only includes the mass, initial speed, final speed, and work done. So, we'll need to use the work-energy theorem to find the change in potential energy.
According to the work-energy theorem:
Net Work = ΔKE + ΔPE
where Net Work is the total work done on an object, ΔKE is the change in kinetic energy, and ΔPE is the change in potential energy.
In this case, the net work done on the skateboarder is the sum of the work done by the skateboarder on himself and the work done by friction:
Net Work = Work by skateboarder on himself + Work by friction
Net Work = 90.8 J + (-278 J) (Remember that the work done by friction is negative because it opposes motion.)
Net Work = -187.2 J
Since no other external work is mentioned, the net work is equal to the change in kinetic energy:
Net Work = ΔKE
ΔKE = -187.2 J
The change in kinetic energy can be calculated using the formula:
ΔKE = (1/2) * m * (vf^2 - vi^2)
where m is the mass, vf is the final velocity, and vi is the initial velocity.
Substituting the known values:
-187.2 J = (1/2) * 60.8 kg * (8.12 m/s)^2 - (1/2) * 60.8 kg * (1.84 m/s)^2
Now, solve the equation to find the final velocity of the skateboarder:
187.2 J = 0.5 * 60.8 kg * (8.12 m/s)^2 - 0.5 * 60.8 kg * (1.84 m/s)^2
187.2 J = 0.5 * 60.8 kg * (66.0944 m^2/s^2) - 0.5 * 60.8 kg * (3.3856 m^2/s^2)
187.2 J = 0.5 * 60.8 kg * 62.7088 m^2/s^2
187.2 J = 3819.2384 kg * m^2/s^2
Therefore, we can conclude that the change in kinetic energy is 3819.2384 kg * m^2/s^2.
Now, let's go back to the formula for the net work done:
Net Work = ΔKE + ΔPE
Rewriting the equation to solve for ΔPE:
ΔPE = Net Work - ΔKE
Substituting the known values:
ΔPE = (-187.2 J) - (3819.2384 kg * m^2/s^2)
ΔPE = -4006.4384 kg * m^2/s^2
Therefore, the change in gravitational potential energy (PEf - PE0) is -4006.4384 kg * m^2/s^2.
(b) Calculate the change in vertical height (Δh):
To find the change in vertical height (Δh), we can rearrange the formula for potential energy:
ΔPE = m * g * h
Rearranging the equation to solve for Δh:
Δh = ΔPE / (m * g)
Substituting the known values:
Δh = (-4006.4384 kg * m^2/s^2) / (60.8 kg * 9.8 m/s^2)
Δh = -6.419 meters
The absolute value of the change in vertical height is 6.419 meters. Therefore, the vertical height of the skater has changed by approximately 6.419 meters.