Describe the motion of a ball being thrown straight up in the air in terms of kinetic and potential energy. as the ball rises up and falls back down, how is its kinetic and potential energy affected? when is each type of energy the greatest and the least? what is true about the total mechanical energy of the ball?

Kinetic energy = energy in motion 
Potential energy = energy possessed by body (not in motion)

There's an equation which states that the Total mechanical energy = Kinetic Energy + Potential Energy 

Kinetic Energy = Total Mechanical Energy - Potential Energy and Potential Energy= Total Mechanical energy - kinetic energy

This makes the relationship clear, that when the ball is in motion...there's more kinetic energy, and when it is coming to a stop or static...there is more potential energy. 

So, using this knowledge you can answer the questions.

1st Question: As the ball leaves your hand, There is only Kinetic Energy...with the potential energy at 0. Then as it starts to slow down due to the effect of Gravity and air resistance, the Kinetic energy gradually starts to drop at the same rate at which the potential energy rises. Once it reaches a complete stop in the Air. The potential energy is at its maximum, with the kinetic at 0. Then as it begins to come back down, the Potential Energy begins to drop at the same rate in which the Kinetic Energy rises. This loop will continue if the ball is not caught and is instead allowed to bounce until it comes to a complete stop. If the ball is caught with your hand, the ball will finish with maximum potential energy and 0 kinetic energy.

2nd Question: I've already answered this in the answer to the first one. But just to rephrase.
Kinetic Energy is Greatest when it leaves the hand and when it begins to come back down.
Potential Energy is Greatest When it is still in your hand and when it reaches a complete stop mid air.

3rd Question: The total mechanical Energy of the Ball is the Kinetic Energy + the Potential Energy. As the Kinetic Energy rises, Potential Energy drops and Vice versa. 

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