There are three forces acting on waste. First, there is downward gravity (Fg) because of the connection with the Earth. This force depends on the mass (m) of the object and the force of gravity (g = 9.8 newtons per kilogram on Earth).
After that, we have an additional force (Fb). When an object is submerged in water (or any liquid), there is an upward force from the surrounding water. The magnitude of this force is proportional to the weight of the water displaced, so it is proportional to the volume of the object. Note that gravity and gravity depend on the size of the object.
Finally, we have gravity (Fd) due to the interaction between the flowing water and the object. This force depends on the size of the object and its speed relative to the water. We can calculate the magnitude of gravity (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 viscosity, and v is the velocity of the fluid relative to the object. In water, the viscosity has a value of 0.89 x 10-3 kilogram per meter per second.
Now we can model the movement of a rock and the movement of a piece of gold in moving water. There is one small issue, however. According to Newton’s second law, the net force on an object changes the object’s velocity—but when the velocity changes, so does the force.
One way to deal with this issue is to break the flow of each item into a short period of time. At any given time, I can assume that the net force is constant (which is almost certainly true). With a constant force, I can find the velocity and position of the object at the end of the interval. Then I just need to repeat the same process for the next time.