Aball is thrown into the air and caught at the exact same distance from the ground. at what point is the ball's kinetic energy the greatest? include air friction in your consideration.
a. just before it is caughtb. at the highest point of its pathc. the instant it leaves the thrower's handd. halfway between where it is thrown and the highest point

Remember that the ball can never have MORE energy than you give it
when you toss it. 

You give the ball kinetic energy.  As it goes up, the kinetic energy
changes to potential energy, and on the way down, that potential
energy changes back into kinetic energy. 

If there's nothing along the way to steal any energy from the ball,
then its TOTAL energy (kinetic PLUS potential) is constant all the
way up and down.

But in this question, there IS something else in the picture, and it
steals energy from the ball.  Air resistance is partly the friction of
the air molecules scraping against the ball, and partly the work
that the ball has to do in order to push air out of the way so that
it can get through.  All of that steals energy whenever the ball is
moving.

So figure it out:  If you give it some energy when you throw it,
and it can never have any more than that because it doesn't
have little jet engines on it, and the air is constantly robbing
some of the energy whenever the ball is moving, then the most
it can ever have is just as it leaves your hand.