Bridge over the Thames at Abingdon: F = -gradV

We were sat in Annie's cafe by the bridge in Abingdon and looked up our height above sea level. It is 120 metres above sea level. This difference is enough to drive all that water about 100 miles down to the sea. I had just been reading that the force acting on something = - potential energy gradient. That can be written as -grad V where grad is a differential operator. Potential energy is a scalar quantity with each point on the landscape having its own value. But the numbers vary from place to place and thus we can say that there are places where the potential energy is going up and places where it is going down if we move across the landscape. Hence the gradient. In one dimension it is the simple differential dV/dx.

Let's say that we have to go distance x down to the sea. There is potential energy mgh where we are and 0 at the sea by definition. The gradient thus becomes mgh/x. As shown below, the component of the gravitational force mg driving the motion is mgsin(theta). Now it works if sin(theta) = h/x, which is true if the distance x is the distance measured down the slope to the sea. It doesn't matter too much over such a large distance because the difference in length between the two long sides will be virtually impossible to measure.