There are three forces acting on the debris. First, there is a pull-down gravity (Fg) due to interaction with Earth. This force depends on both the mass (m) of the object and the gravitational field (g = 9.8 newtons/kg on Earth).
Next, we have the buoyancy 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 equal to the weight of the displaced water, so that it is proportional to the volume of the object. Note that both gravity and buoyancy depend on the size of the object.
Finally, we have the drag force (FD) due to the force of 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 drag (in water, not to be confused with balloon string) use Stokes’ Lawaccording to the following formula:
In this expression, R is the radius of the spherical body, μ is the kinematic viscosity, and v is the velocity of the fluid relative to the object. In water, the kinematic viscosity is about 0.89 x 10 .-3 kilograms per meter per second.
We can now model the motion of a stone relative to the motion of a piece of gold in a moving stream of water. There is one small problem, though. Based on Newton’s Second LawThe sum of the forces acting on an object changes the object’s velocity—but as the velocity changes, so does the force.
One way to solve this problem is to divide the motion of each object into small time intervals. For each period, I can assume that the net force is constant (which is approximate). With a constant force I can then find the velocity and position of the object at the end of that interval. Then I just repeat the same process for the next interval.