A young kid with a huge baseball cap is playing catch with himself by throwing a ball straight up. How fast do

A young kid with a huge baseball cap is playing catch with himself by throwing a ball straight up. How fast does he throw it if the ball comes back to his hands 0.8 seconds later?

Since it comes back to his hands .8 seconds later, and assuming that he catches the ball with his hand in the same position as he was when he threw it, you can say that it took half the time for the ball to reach the apex, or very top of the flight, or .4 seconds.

The velocity at the apex (final velocity) is 0 m/s.

Knowing these things, you can put them into the equation: Vf = Vi + at
Vf = final velocity, Vi = initial velocity (or throwing speed), a = acceleration, t = time
Vf = 0 m/s, a = -9.8 m/s^2 (negative because it's going against gravity), t = .4 s
Solve for Vi.

Vf = Vi + at
0 = Vi + (-9.8)(.4)
0 = Vi + (-39.2)
39.2 = Vi

He threw the ball at a velocity of 39.2 m/s.

P.S. Does the size of the baseball cap matter? :P