A child and sled with a combined mass of 58.9 kg slide down a frictionless hill that is 8.09 m high at an angle of 23◦ from horizontal.

The acceleration of gravity is 9.81 m/s2 .
If the sled starts from rest, what is its speed at the bottom of the hill?
Answer in units of m/s

To find the speed of the sled at the bottom of the hill, we can use the principle of conservation of mechanical energy.

1. Determine the potential energy at the top of the hill:
The potential energy (PE) at the top of the hill can be calculated using the formula:
PE = mass * gravity * height
where mass is the total mass of the child and sled (58.9 kg), gravity is the acceleration due to gravity (9.81 m/s^2), and height is the height of the hill (8.09 m).
PE = (58.9 kg) * (9.81 m/s^2) * (8.09 m)

2. Calculate the kinetic energy at the bottom of the hill:
Since the sled starts from rest, its initial kinetic energy (KE) is zero. At the bottom of the hill, all the potential energy is converted into kinetic energy.
Therefore, KE = PE

3. Calculate the speed at the bottom of the hill:
The kinetic energy (KE) can be calculated using the formula:
KE = (1/2) * mass * velocity^2
where mass is the total mass of the child and sled (58.9 kg), and velocity is the speed of the sled at the bottom of the hill.
(1/2) * (58.9 kg) * velocity^2 = (58.9 kg) * (9.81 m/s^2) * (8.09 m)

4. Solve for velocity:
Simplifying the equation, we have:
(1/2) * velocity^2 = (9.81 m/s^2) * (8.09 m)
velocity^2 = 2 * (9.81 m/s^2) * (8.09 m)
velocity^2 = 159.964
velocity ≈ 12.65 m/s

Therefore, the speed of the sled at the bottom of the hill is approximately 12.65 m/s.