Work and Energy - High diving into a pool

Problem Statement:

A diver jumps from a height h with an angle α0 to the horizontal. The magnitude of his initial velocity is v0. Use the energy conservation principle to determine the maximum height reached by the diver and his speed when he reaches the water surface (we will neglect the friction of the air). Do these two magnitudes depend on α0?

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Solution:

Since we neglect the friction of the air, the only force that acts on the diver during his motion is his weight. As a result, the mechanical energy is conserved.

To solve the problem we will compare the mechanical energy of the diver at the initial state A (when he jumps) with his energy at the highest point of trajectory B. The trajectory is a parabola, as seen in the following figure. The kinetic energy is not zero at point B, since the jumper has a velocity (horizontal) at this point.

The speed of the diver at point B is that which corresponds to a body that describes a parabolic motion at the highest point of its trajectory:

By imposing the principle of conservation of energy between points A and B we get:

Which allows us to isolate h0: