A child and sled with a combined mass of 45.0 kg slide down a frictionless hill that is 8.79 m high. If the sled starts from rest, what is its speed at the bottom of the hill?

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.

The initial potential energy (PE) of the sled and child at the top of the hill is equal to the final kinetic energy (KE) of the sled at the bottom of the hill, assuming no energy losses due to friction.

PE(top) = KE(bottom)

The potential energy at the top of the hill can be calculated using the formula:

PE = m * g * h

Where:
m = mass of the sled and child (45.0 kg)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height of the hill (8.79 m)

Substituting the values:

PE(top) = 45.0 kg * 9.8 m/s^2 * 8.79 m

Now, let's calculate the value of potential energy at the top:

PE(top) = 3925.77 J

Since the potential energy at the top is equal to the kinetic energy at the bottom, we can set the two values equal to each other:

PE(top) = KE(bottom)

KE(bottom) = 3925.77 J

The kinetic energy formula is given by:

KE = (1/2) * m * v^2

Where:
m = mass of the sled and child (45.0 kg)
v = velocity of the sled at the bottom of the hill

Now, let's substitute the values:

3925.77 J = (1/2) * 45.0 kg * v^2

Simplifying the equation:

784.15 = 22.5 * v^2

Dividing both sides by 22.5:

v^2 = 34.84

Taking the square root of both sides:

v ≈ 5.9 m/s

Therefore, the speed of the sled at the bottom of the hill is approximately 5.9 m/s.