I have a question that reads as the following:an object is dropped from the top of a high building and falls under the force of gravity alone. Which option gives the graph of the potential energy U of the object as a function of the distance x through which it has fallen?

My first instinct was for it to be a graph curving towards the x axis(like a half a rainbow)

But another one that seems feasible is a graph worth a straight line from the U(x) axis down onto the x axis

Any help please!!!

Well, the gravitational potential energy function (near the surface of the Earth) is

\(\displaystyle U = -mgh\)

so it's a straight line with a negative slope. (The red line in the graph below.)

If we are talking about Newtonian gravity then we have

\(\displaystyle U = -\frac{GM_Em}{R_E + h}\)

where \(\displaystyle M_E \text{ and } R_E\) are the mass and radius of the Earth, respectively. This looks like the blue curve in the graph below.

-Dan