An 91.0 kg in-line skater does 3620 J of nonconservative work by pushing against the ground with his skates. In addition, friction does -615 J of nonconservative work on the skater. The skater's initial and final speeds are 2.70 m/s and 1.02 m/s, respectively.
Part B
Calculate the change in height of the skater.
Ke initial = (1/2)(91)(2.7)^2 = 332 Joules
Ke final = (1/2)(91)(1.02)^2 = 47.3 Joules
332 J - 615 J + 3620 J = 47.3 J + m g h
so
m g h = 3290 Joules
h = 3290/ (91*9.81) = 3.69 meters
(how did the ice get tilted ? :)
To calculate the change in height of the skater, we need to determine the total mechanical energy of the system, which includes both the potential energy (height) and the kinetic energy (speed) of the skater.
The total mechanical energy, E, is given by the equation:
E = KE + PE
where KE is the kinetic energy and PE is the potential energy.
In this case, we are given the initial and final speeds of the skater, so we can calculate the change in kinetic energy:
ΔKE = KE_final - KE_initial
Substituting the given values:
ΔKE = (1/2) * m * v_final^2 - (1/2) * m * v_initial^2
where m is the mass of the skater (91.0 kg), v_final is the final speed (1.02 m/s), and v_initial is the initial speed (2.70 m/s).
ΔKE = (1/2) * 91.0 kg * (1.02 m/s)^2 - (1/2) * 91.0 kg * (2.70 m/s)^2
Next, we need to calculate the work done by the skater and the work done by friction.
Work done by the skater, W_skater = 3620 J
Work done by friction, W_friction = -615 J
The work-energy principle states that the work done on an object is equal to the change in its mechanical energy:
W_net = ΔKE + ΔPE
where W_net is the net work done on the system and ΔPE is the change in potential energy (height).
Since we know the net work done on the system is equal to the sum of the work done by the skater and the work done by friction:
W_net = W_skater + W_friction
Substituting the given values:
W_net = 3620 J + (-615 J)
Now we can substitute the values into the equation to solve for the change in potential energy:
3620 J - 615 J = (1/2) * 91.0 kg * (1.02 m/s)^2 - (1/2) * 91.0 kg * (2.70 m/s)^2 + ΔPE
Simplifying, we get:
3005 J = (1/2) * 91.0 kg * (1.02 m/s)^2 - (1/2) * 91.0 kg * (2.70 m/s)^2 + ΔPE
3005 J = 46.8516 kg·m^2/s^2 - 134.5515 kg·m^2/s^2 + ΔPE
3005 J = -87.6999 kg·m^2/s^2 + ΔPE
To find the change in potential energy, we can rearrange the equation:
ΔPE = 3005 J + 87.6999 kg·m^2/s^2
Finally, we can calculate the change in height by using the formula for potential energy:
ΔPE = m * g * Δh
where m is the mass of the skater (91.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and Δh is the change in height (in meters).
Rearranging the equation, we get:
Δh = ΔPE / (m * g)
Substituting the values, we find:
Δh = (3005 J + 87.6999 kg·m^2/s^2) / (91.0 kg * 9.8 m/s^2)
Solving this equation will give us the change in height of the skater.