A child and sled with a combined mass of 45.0 kg slide down a frictionless hill. If the sled starts from rest and has a speed of 8.0 m/s at the bottom, what is the height of the hill (include units)?
PE = KE
Mg*h = 0.5M*V^2.
M*g = 45*9.8.
h = ?
To find the height of the hill, we can use the principle of conservation of mechanical energy. The total mechanical energy at the top of the hill equal to the total mechanical energy at the bottom.
The total mechanical energy is the sum of the potential energy and the kinetic energy.
At the top of the hill, the sled is at rest, so the kinetic energy is zero. Therefore, the total mechanical energy at the top is equal to the potential energy.
At the bottom of the hill, the sled has a speed of 8.0 m/s. Therefore, the total mechanical energy at the bottom is equal to the sum of the potential energy and the kinetic energy.
Let's assume the height of the hill is h.
At the top of the hill:
Potential energy = mgh
At the bottom of the hill:
Potential energy + Kinetic energy = mgh + 1/2 mv^2
Since the hill is frictionless, there is no loss of mechanical energy between the top and the bottom.
Therefore, we have:
mgh = mgh + 1/2 mv^2
Canceling out the mass 'm' on both sides, we get:
gh = gh + 1/2 v^2
Rearranging the equation to solve for 'h':
gh - gh = 1/2 v^2
gh - gh = 1/2 (8.0 m/s)^2
h(g - g) = 1/2 (64 m^2/s^2)
h = 1/2 (64 m^2/s^2) / g
Using the acceleration due to gravity, g = 9.8 m/s^2, we can substitute in the values:
h = 1/2 (64 m^2/s^2) / 9.8 m/s^2
h ≈ 3.27 meters
Therefore, the height of the hill is approximately 3.27 meters.
To determine the height of the hill, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of an object moving along a frictionless path remains constant.
The total mechanical energy of an object can be calculated as the sum of its kinetic energy (KE) and potential energy (PE):
E = KE + PE
At the top of the hill, the sled is at rest, and its kinetic energy is zero since its velocity is zero. Therefore, at the top of the hill, all of the energy is in the form of potential energy, given by:
PE_top = m * g * h
Where m is the mass of the sled and child (45.0 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the hill we want to find.
At the bottom of the hill, all of the energy is in the form of kinetic energy:
KE_bottom = 1/2 * m * v^2
Where v is the velocity of the sled at the bottom of the hill (8.0 m/s).
Since the total mechanical energy is conserved, we can set the potential energy at the top of the hill equal to the kinetic energy at the bottom of the hill:
PE_top = KE_bottom
m * g * h = 1/2 * m * v^2
Canceling out the mass:
g * h = 1/2 * v^2
Simplifying the equation:
h = (1/2) * (v^2) / g
Substituting the known values:
h = (1/2) * (8.0^2) / 9.8
Calculating the result:
h = 32 / 9.8
h ≈ 3.27 meters
Therefore, the height of the hill is approximately 3.27 meters.