I threw a rock straight up with a speed of 13 m/s. I happened to be standing over a well when I threw it, so when instead of landing on the ground,...

Hello!

Let's count a height from the level of the water surface. Then the height of the water surface is zero and the height of your hand is Also we'll ignore air resistance.

The height of a rock over the water surface is described by the formula

where is the initial speed. This is the projection on the upward axis, therefore has plus sign and has minus sign.

Our task is to find the time at which Of course must be greater than zero.

In numbers, the equation is  

the positive solution is  

This is the answer.

[The negative solution (with the minus before the root) has some sense. It corresponds to the imaginary situation when you would throw a rock from the water level. Then that time (its absolute value) is how long it would take for a rock to reach the height of your hand.]