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 conservation of energy. The mass won't matter; it cancels out of the energy conservation equaiton.
M g H = (1/2) M V^2
Solve for H, in terms of V.
To find the height of the hill, we can use the principle of conservation of mechanical energy. The total mechanical energy (E) of the system (child + sled) is conserved, which means that it remains constant throughout the slide.
The total mechanical energy of an object can be given by the sum of its kinetic energy (KE) and potential energy (PE).
KE = 1/2 * mass * velocity^2
PE = mass * gravity * height
In this case, since the sled starts from rest at the top of the hill, its initial kinetic energy is zero. Therefore, at the top of the hill, the total mechanical energy is equal to the potential energy:
E = PE = mass * gravity * height
At the bottom of the hill, the total mechanical energy is the sum of the kinetic energy and potential energy:
E = KE + PE = 1/2 * mass * velocity^2 + mass * gravity * height
Since the mass of the child and sled combined is given as 40.0 kg, and the velocity at the bottom is given as 9.0 m/s, we can substitute these values into the equation:
E = 1/2 * 40.0 kg * (9.0 m/s)^2 + 40.0 kg * gravity * height
Now, we know that the acceleration due to gravity is approximately 9.8 m/s^2. Substituting this value and rearranging the equation, we can solve for the height:
height = (E - 1/2 * 40.0 kg * (9.0 m/s)^2) / (40.0 kg * gravity)
Substituting the known values, we can calculate the height of the hill.