A 61.8 kg skier coasts up a snow-covered hill that makes an angle of 26.0° with the horizontal. The initial speed of the skier is 6.10 m/s. After coasting a distance of 1.86 m up the slope, the speed of the skier is 4.48 m/s. Calculate the work done by the kinetic frictional force that acts on the skis.
intialKE-changePE-friction=finalKE
for the change in PE, use the slope distance and sine of the angle to get change in height.
To calculate the work done by the kinetic frictional force, we need to first calculate the change in kinetic energy of the skier.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the kinetic frictional force will be equal to the change in the skier's kinetic energy.
The formula for calculating the work done is:
Work = (Final Kinetic Energy) - (Initial Kinetic Energy)
To calculate the Initial Kinetic Energy, we use the formula:
Initial Kinetic Energy = (1/2) * mass * (Initial Velocity)^2
To calculate the Final Kinetic Energy, we use the formula:
Final Kinetic Energy = (1/2) * mass * (Final Velocity)^2
Given:
Mass (m) = 61.8 kg
Angle (θ) = 26.0°
Initial Velocity (V₀) = 6.10 m/s
Final Velocity (V) = 4.48 m/s
Distance (s) = 1.86 m
First, we need to calculate the change in height (Δh) using the distance and the angle of the slope:
Δh = s * sin(θ)
Now, let's calculate the initial and final heights:
Initial height (h₀) = 0 m (since the skier starts at ground level)
Final height (h) = Δh
Next, we can calculate the gravitational potential energy difference:
ΔPE = m * g * (h - h₀)
Where g is the acceleration due to gravity (approximately 9.8 m/s²).
Finally, we can calculate the change in kinetic energy using the formulas mentioned earlier:
Initial Kinetic Energy = (1/2) * m * (V₀)²
Final Kinetic Energy = (1/2) * m * (V)²
The work done by the kinetic frictional force will be equal to the change in kinetic energy:
Work = Final Kinetic Energy - Initial Kinetic Energy
Calculating each step:
Δh = 1.86 m * sin(26.0°) = 0.798 m
ΔPE = 61.8 kg * 9.8 m/s² * (0.798 m - 0 m) = 482.2596 J
Initial Kinetic Energy = (1/2) * 61.8 kg * (6.10 m/s)² = 1137.33 J
Final Kinetic Energy = (1/2) * 61.8 kg * (4.48 m/s)² = 722.3168 J
Work = 722.3168 J - 1137.33 J = -415.0132 J
The negative sign indicates that work is done against the motion of the skier, which is consistent with the work done by friction.
Therefore, the work done by the kinetic frictional force that acts on the skis is approximately -415.0132 Joules.