A child and sled with a combined mass of 53.0 kg slide down a frictionless hill that is 7.63 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
gh = 1/2 v^2
v = √2gh = √(2*9.8*7.63) = 12.23 m/s
note that the mass doesn't matter. Only the height contributes to the speed.
To find the speed of the sled at the bottom of the hill, we can use the principle of conservation of energy. The gravitational potential energy (GPE) at the top of the hill is converted into kinetic energy (KE) at the bottom of the hill.
The gravitational potential energy (GPE) at the top of the hill can be calculated using the formula:
GPE = mass × gravity × height
where mass = 53.0 kg, gravity = 9.8 m/s^2 (acceleration due to gravity), and height = 7.63 m.
Thus, GPE = 53.0 kg × 9.8 m/s^2 × 7.63 m = 3949.46 J
According to the principle of conservation of energy, the GPE is converted to kinetic energy (KE) at the bottom of the hill. The formula for kinetic energy is:
KE = (1/2) × mass × velocity^2
where mass = 53.0 kg and velocity is the speed of the sled at the bottom.
Equating the GPE to the KE:
GPE = KE
3949.46 J = (1/2) × 53.0 kg × velocity^2
Simplifying the equation:
3949.46 J = 26.5 kg × velocity^2
Dividing both sides by 26.5 kg:
velocity^2 = 3949.46 J / 26.5 kg
velocity^2 ≈ 149 m^2/s^2
To find the velocity, we take the square root of both sides:
velocity ≈ √(149 m^2/s^2) ≈ 12.19 m/s
Therefore, the speed of the sled at the bottom of the hill is approximately 12.19 m/s.