A child and sled with a combined mass of 40.0 kg slide down a frictionless hill. If the sled starts from rest and has a speed of 9.0 m/s at the bottom, what is the height of the hill?
m
Use energy:
gravitational potential energy: GPE = mgh
kinetic energy: KE = (1/2)m*v^2
By conservation of energy, GPEi+KEi = GPEf + KEf. (Final energy = initial energy).
At the bottom of the hill, GPEf = 0 (h=0). At the top of the hill, KEi = 0(v=0). Find the final kinetic energy, and from it, the final velocity.
To find the height of the hill, we can use the principle of conservation of energy. The potential energy at the top of the hill is equal to the sum of the kinetic energy at the bottom and the work done by gravity.
The potential energy (PE) at the top of the hill can be calculated using the formula PE = mgh, where m is the mass (40.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the hill (unknown).
The kinetic energy (KE) at the bottom of the hill can be calculated using the formula KE = 1/2mv^2, where m is the mass (40.0 kg), and v is the velocity (9.0 m/s).
Since the hill is frictionless, there is no work done against friction, so the total mechanical energy is conserved.
Let's set up the equation:
PE (at the top) = KE (at the bottom)
mgh = 1/2mv^2
Canceling out the mass term, we have:
gh = 1/2v^2
Solving for h, we get:
h = (1/2v^2) / g
Plugging in the values, we have:
h = (1/2 * 9.0^2) / 9.8
h = (1/2 * 81) / 9.8
h = 40.5 / 9.8
h ≈ 4.13 m
Therefore, the height of the hill is approximately 4.13 meters.