The Physics of ‘Sniping’ for Gold

There are three forces acting on the debris. First, there’s the downward-pulling gravitational force (F_g) due to the 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).

Next, we have the buoyancy force (F_b). When an object is submerged in water (or any fluid), there is an upward-pushing force from the surrounding water. The magnitude of this force is equal to the weight of the water displaced, such that it’s proportional to the volume of the object. Notice that both the gravitational force and the buoyancy force depend on the size of the object.

Finally, we have a drag force (F_d) 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 air drag) 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 with respect 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 versus the motion of a piece of gold in moving water. There is one small issue, though. According to Newton’s second law, the net force on an object changes the object’s velocity—but as the velocity changes, the force also changes.

One way to deal with this issue is to break the motion 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 velocity and position of the object at the end of the interval. Then I just need to repeat this same process for the next interval.