A 20.0 kg cart, initially at rest, rolls down a hill. What is the speed of the cart when it is 5.00 meters lower than its initial position?

To find the speed of the cart when it is 5.00 meters lower than its initial position, you can use the principle of conservation of energy. The potential energy lost by the cart as it moves down the hill is converted into kinetic energy, which determines its speed.

First, calculate the potential energy lost by the cart as it moves 5.00 meters lower:

Potential energy lost = mgh

Where: m = mass of the cart (20.0 kg) g = acceleration due to gravity (approximately 9.8 m/s^2) h = height difference (5.00 meters)

Next, calculate the kinetic energy gained by the cart:

Kinetic energy gained = 1/2 * mv^2

Where: m = mass of the cart (20.0 kg) v = speed of the cart (to be determined)

According to the conservation of energy principle, the potential energy lost equals the kinetic energy gained:

mgh = 1/2 * mv^2

Now, solve for the speed (v):

v = √(2gh)

Substitute the known values:

v = √(2 * 9.8 m/s^2 * 5.00 m)

Calculate:

v = √(98 m^2/s^2)

v ≈ 9.90 m/s

Therefore, the speed of the cart when it is 5.00 meters lower than its initial position is approximately 9.90 meters per second.