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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 15.5 m/s via path 1. How fast is he moving on path 2 when he releases the rope at a height of 4.30 m above the water? Ignore the effects of air resistance.

To answer this question, we can make use of the principle of conservation of mechanical energy. The total mechanical energy of the person at the top of the cliff is equal to the sum of potential energy and kinetic energy, and this total mechanical energy should remain unchanged throughout the motion.

Let's denote the potential energy at the top of the cliff as PE1, the kinetic energy when the person releases the rope on path 1 as KE1, and the potential energy when the person releases the rope on path 2 as PE2.

Initially, at the top of the cliff, the person has only potential energy, which can be expressed as:

PE1 = m * g * h1

where m is the mass of the person, g is the acceleration due to gravity, and h1 is the height of the cliff.

When the person reaches path 1 and releases the rope, all potential energy is converted into kinetic energy. So, we can write:

KE1 = PE1

Next, when the person reaches path 2, the kinetic energy is converted back into potential energy due to the decrease in height. At this point, the person has only potential energy:

PE2 = m * g * h2

where h2 is the height above the water when the person releases the rope on path 2.

Since the total mechanical energy remains constant, we can equate the two expressions for the total mechanical energy:

PE1 + KE1 = PE2

Substituting the respective expressions, we get:

m * g * h1 + KE1 = m * g * h2

Simplifying the equation, we find:

KE1 = m * g * (h2 - h1)

Now, we can use this equation to find the value of KE1, which represents the kinetic energy when the person releases the rope on path 1.

Finally, we can calculate the speed at which the person is moving on path 2 when they release the rope using the formula for kinetic energy:

KE2 = 1/2 * m * v^2

where KE2 is the kinetic energy at path 2, and v is the speed on path 2.

Rearranging the equation, we can solve for v:

v = sqrt(2 * KE2 / m)

Substituting the value of KE2 that we determined earlier, we can find the speed at which the person is moving on path 2 when they release the rope.