Could someone please explain how you get this answer???
A skier with a mass of 60.0 kg pushes off with a horizontal speed of 3.0 m/s from the top of a snow covered hill. As he slides down the hill, his height decreases by 42 m over a distance of 100 m of travel. How much will his kinetic energy increase by the time he reaches the bottom of the hill (assuming no friction)?
Change in Ke is equal to the loss of potential energy
m g h = 60 * 9.81 * 42 Joules
To calculate the change in kinetic energy of the skier as he slides down the hill, we need to use the principles of work and energy. The change in kinetic energy can be determined by finding the difference between the initial and final kinetic energy of the skier.
1. Calculate the initial kinetic energy:
The initial kinetic energy (KEi) can be calculated using the formula:
KEi = 1/2 * mass * (velocity)^2
Given:
mass (m) = 60.0 kg
Initial velocity (vi) = 3.0 m/s
Plugging in the values, we have:
KEi = 1/2 * 60.0 kg * (3.0 m/s)^2
KEi = 270.0 J
So, the initial kinetic energy of the skier is 270.0 J.
2. Calculate the final kinetic energy:
The final kinetic energy (KEf) can be calculated using the formula:
KEf = 1/2 * mass * (velocity)^2
Given:
Final velocity (vf) - As the skier reaches the bottom of the hill, the final velocity is not given, but we can calculate it using the concept of conservation of energy. Since there is no friction involved, the potential energy the skier loses while descending the hill will be converted to kinetic energy.
Change in height (Δh) = -42 m (negative sign indicates decrease in height)
Distance traveled (d) = 100 m
Using the concept of conservation of energy, we can equate the change in potential energy to the change in kinetic energy:
m * g * Δh = 1/2 * m * (vf)^2
where g is the gravitational acceleration (9.8 m/s^2)
Solving for vf:
vf^2 = 2gΔh
vf^2 = 2 * 9.8 m/s^2 * 42 m
vf^2 = 823.2 m^2/s^2
vf ≈ 28.7 m/s
Plugging in the values, we have:
KEf = 1/2 * 60.0 kg * (28.7 m/s)^2
KEf ≈ 24,771.3 J
So, the final kinetic energy of the skier is approximately 24,771.3 J.
3. Calculate the change in kinetic energy:
The change in kinetic energy (ΔKE) can be calculated by subtracting the initial kinetic energy from the final kinetic energy:
ΔKE = KEf - KEi
Plugging in the values, we have:
ΔKE = 24,771.3 J - 270.0 J
ΔKE ≈ 24,501.3 J
Therefore, the skier's kinetic energy will increase by approximately 24,501.3 Joules by the time he reaches the bottom of the hill (assuming no friction).