The drawing below shows a person who, starting from rest at the top of a cliff, swings down at the end of a rope, releases it, and falls into the water below. There are two paths by which the person can enter the water. Suppose he enters the water at a speed of 18.5 m/s via path 1. How fast is he moving on path 2 when he releases the rope at a height of 4.00 m above the water? Ignore the effects of air resistance.

To find the speed of the person on path 2 when he releases the rope, we can use the principle of conservation of energy. We'll start by calculating his initial potential energy at the top of the cliff and equating it to his final kinetic energy just before releasing the rope.

First, let's find the potential energy at the top of the cliff (path 1) using the equation:

Potential Energy = mass × gravity × height

Since the mass of the person is not given, we can cancel it out when comparing the potential energies, so we'll simply calculate the ratio of heights:

Potential Energy at top of cliff (Path 1) / Potential Energy at height 4.00 m (Path 2) = Height at top of cliff / Height at height 4.00 m

Potential Energy at top of cliff (Path 1) / Potential Energy at height 4.00 m (Path 2) = 0 m / 4.00 m (since the person starts from rest at the top of the cliff, its height is 0)

Potential Energy at top of cliff (Path 1) / Potential Energy at height 4.00 m (Path 2) = 0 / 4.00 = 0

Since the potential energy ratio is 0, it means the person has no potential energy left on path 2 when he releases the rope. Therefore, all of the energy is converted into kinetic energy.

Using the equation for kinetic energy:

Kinetic Energy = (1/2) × mass × velocity^2

Since we know the person's speed on path 1 is 18.5 m/s, we can set up the equation:

(1/2) × mass × velocity^2 = 0.5 × mass × (18.5)^2

Next, we need to solve for the velocity on path 2:

velocity on path 2 = √(velocity^2 on path 1 - (2 × gravity × height))

Substituting the given values:

velocity on path 2 = √((18.5)^2 - (2 × 9.8 × 4.00))

Calculating this expression will give us the speed at which the person is moving on path 2 when he releases the rope.