A 60.0-kg skier slides down a hill. She starts with a speed of 0.50 m/s. The friction force is only 10 N. The slope is 80 m long and 50 m high. What is the speed of the skier at the bottom of the hill?
M*g = 60*9.8 = 588 N. = Wt. of skier.
PE = 0.5M*V^2 + Fki*d = 588h.
30V^2 + 10*80 =588*50,
V = 30.9 m/s.
To find the speed of the skier at the bottom of the hill, we can use the principle of conservation of mechanical energy.
The initial energy of the skier at the top of the hill consists of two components: the gravitational potential energy and the kinetic energy.
The gravitational potential energy is given by the formula:
Potential Energy = mass * gravitational acceleration * height
In this case, the mass of the skier is 60.0 kg, the gravitational acceleration is approximately 9.8 m/s^2, and the height of the hill is 50 m. Plugging in these values, we find:
Potential Energy = 60.0 kg * 9.8 m/s^2 * 50 m = 29400 J
The initial kinetic energy of the skier is given by the formula:
Kinetic Energy = (1/2) * mass * velocity^2
In this case, the mass of the skier is 60.0 kg and the initial velocity is 0.50 m/s. Plugging in these values, we find:
Kinetic Energy = (1/2) * 60.0 kg * (0.50 m/s)^2 = 7.5 J
The total initial energy of the skier is the sum of potential and kinetic energy:
Total Energy = Potential Energy + Kinetic Energy = 29400 J + 7.5 J = 29407.5 J
At the bottom of the hill, the potential energy is zero because the skier is at ground level. Therefore, the entire energy of the skier is in the form of kinetic energy.
We know that the work done by friction (F_friction) is equal to the force applied (F_friction) multiplied by the distance traveled (d). In this case, the friction force is 10 N and the distance traveled is 80 m. Therefore, the work done by friction is:
Work Friction = F_friction * d = 10 N * 80 m = 800 J
According to the principle of conservation of mechanical energy, the total energy at the bottom of the hill should be equal to the initial total energy minus the work done by friction:
Total Energy (at the bottom) = Total Energy (initial) - Work Friction
Total Energy (at the bottom) = 29407.5 J - 800 J = 28607.5 J
Since the total energy at the bottom is in the form of kinetic energy, we can use the formula for kinetic energy to find the final velocity of the skier:
Final Velocity^2 = (2 * Total Energy) / mass
Final Velocity^2 = 2 * 28607.5 J / 60.0 kg
Final Velocity^2 = 954.25 J/kg
Finally, taking the square root of the equation, we find:
Final Velocity = √954.25 J/kg ≈ 30 J/kg
The speed of the skier at the bottom of the hill is approximately 30 m/s.