A child and sled with a combined mass of 47.4 kg slide down a frictionless hill that is 8.61 m high. If the sled starts from rest, what is its speed at the bottom of the hill?
PE -> KE
mgh = 1/2 mv^2
Note that the mass makes no difference; only the height.
gh = 1/2 v^2
To find the speed of the sled at the bottom of the hill, we can use the principle of conservation of energy.
The initial potential energy of the sled at the top of the hill is converted into kinetic energy at the bottom, neglecting any losses due to friction or air resistance.
We can start by calculating the initial potential energy (PE) and final kinetic energy (KE).
The formula for potential energy is given by:
PE = m * g * h
Where:
m is the mass of the sled + child (47.4 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height of the hill (8.61 m)
Substituting the given values into the formula:
PE = 47.4 kg * 9.8 m/s^2 * 8.61 m
PE = 3866.46 J
Now, according to the principle of conservation of energy, the initial potential energy (PE) at the top is equal to the final kinetic energy (KE) at the bottom. So,
PE = KE
3866.46 J = 1/2 * m * v^2
Where:
m is the mass of the sled + child (47.4 kg)
v is the velocity of the sled at the bottom (what we're trying to find)
Rearranging the equation to solve for velocity (v):
v^2 = (2 * PE) / m
v^2 = (2 * 3866.46 J) / 47.4 kg
v^2 = 81.721 J/kg
Taking the square root of both sides:
v ≈ √(81.721 J/kg)
v ≈ 9.04 m/s
Therefore, the speed of the sled at the bottom of the hill is approximately 9.04 m/s.