If a ball is thrown vertically upward and we want to calculate the work due to gravity, do we use in the formula the force applied by our hands on...

Hello!

The work due to gravity is the work done by the gravity force. So to find this work we should use gravity force as the force in the formula `W = F*S*cos(alpha).`

The other terms in this formula are the displacement `S` and the angle `alpha` between the force `F` and the displacement `S` (both the force and the displacement are vectors and have a direction).

In the specific case of throwing upwards, `alpha` is equal to `pi` while a ball goes up, and its cosine is `-1.` So the gravity force performs negative work on a ball. While a ball goes down (this happens sooner or later), `alpha=0` and `cos(alpha)=1,` and work becomes positive.

Actually, one may raise the question about the work of the force applied to the ball by our hand. It that case we have to consider that force and take into account that it isn't constant.