LADDER

While the ladder moves from the point A to the point A' where it leaves the wall (Fig.1), the Force Fx (through x-axis) changes from F to 0. That is the only force accelerating the ladder through x-axis, in other words the ladder will reach the maximum speed on x-axis at the time of leave (Fig.2).

The center of mass moves in a circular path in respect of origin O, thus the velocity vector of center is perpendicular to the line from origin to center.

What we are looking for is a value for alpha' that will maximize the Vx, therefore we take the derivative of both sides of the equation above with respect to sin(alpha').

Right side must be zero since the derivative of Vx has to give zero value at the point Vx=Vmax.