forces - Work done against gravity

The work done against gravity is $mgh$, well at least that's what my textbook says. I have a question: I can apply a force say 50N, so total work done = $mgh + mah$. Where $ma$ = Force. But the truth is irrespective of the force applied, the work done against gravity is always $mgh$. Why?

For example, when I move an object with a force, the work done is more, so work depends on the Force. But in case of gravity it always depends upon the weight

Answer

If I take a mass $m$ and apply a force $F$ (greater than $mg$) to it for a distance $h$ upwards then I will do work of:

$$W = Fh \tag{1}$$

The force $F$ has to be greater than the force due to gravity, $mg$, or the object won't move upwards, so let's write the force I apply to the mass as:

$$F = mg + F'$$

then equation (1) becomes:

$$\begin{align} W &= (mg + F')h \\ &= mgh + F'h \end{align}$$

and the first term $mgh$ is work done against gravity while the second term is the work done to increase the velocity of the mass i.e. after the distance $h$ the velocity of the object will be given by:

$$\tfrac{1}{2}mv^2 = F'h$$

or:

$$v = \sqrt{\frac{2F'h}{m}}$$

So if you apply any force $F$ over a distance $h$ then subtract off the increase in the kinetic energy you'll be left with an amount of energy equal to $mgh$. That's why the work done against gravity is always $mgh$.