A ball is thrown straight up from the ground. How high will it go?
The height of the ball at time is , and the force acting on the ball is .
Newton's second law gives the relation
Let for convenience.
Integrate the last equation.
To get , start with an initial velocity and set
Since we have set ,
The equation of position is
To find the maximum, set the first derivative equal to 0.
Check that the second derivative at this point is negative.
So the answer is this point evaluated in the position function.