# The physics of “sniping” for gold

Three forces act on the debris. First, there is the gravitational force that pulls down (Fg) due to interaction with the Earth. This force depends on both the mass (m) of the object and the gravitational field (g = 9.8 newtons per kilogram on Earth).

Then we have the buoyant force (Fb). When an object is immersed in water (or any fluid), there is an upward buoyant force from the surrounding water. The magnitude of this force is equal to the weight of the water displaced, so it is proportional to the volume of the object. Note that the gravitational force and the buoyant force depend on the size of the object.

Finally, we have a drag force (Fd) due to the interaction between the moving water and the object. This force depends on both the size of the object and its relative speed with respect to the water. We can model the magnitude of the drag force (in water, not to be confused with drag in air) using Stoke’s law, according to the following equation:

In this expression, R is the radius of the spherical object, μ is the dynamic viscosity, and v is the velocity of the fluid relative to the object. In water, the dynamic viscosity has a value of about 0.89 x 10-3 kilograms per meter per second.

Now we can model the motion of a rock relative to the motion of a piece of gold in moving water. However, there is a small problem. According to Newton’s second law, the net force on an object changes the speed of the object, but as the speed changes, the force also changes.

One way to deal with this problem is to divide the movement of each object into small time intervals. During each interval I can assume that the net force is constant (which is approximately true). With a constant force, I can then find the speed and position of the object at the end of the interval. Then I just need to repeat this same process for the next interval.