Drag and Gravity Forces on a Falling Sphere
This Demonstration shows the velocity and position over time of two small spheres dropped in air subject to the forces of gravity and drag. You can vary the base mass and surface area of the blue sphere. You can vary the red sphere's mass and area relative to the blue sphere. The differential equation of the forces is described in the Details.
The snapshots show three variations of base and ratio parameters leading to different terminal velocities. The position and velocity of the spheres were directly derived from solutions to the equations , where represents the downward force. Newton's second law gives , where is the instantaneous velocity, is the cross-sectional area, and is the drag coefficient. The drag force acts upward and is proportional to the area and the square of the velocity of the falling sphere. The gravity force acts downward and is proportional to the mass of the sphere; is the acceleration due to gravity. The units are MKS; the mass and area of the sphere scale to roughly model falling raindrops for their terminal velocity. Adjustments in the program could model other shapes and sizes.
The solution with initial condition can be written , where , representing the terminal velocity Integration of ) then gives the distance fallen: .
Linear drag resistance, proportional to the velocity, as described by the Stokes formula, would be valid for Reynolds number . The quadratic drag equation, attributed to Lord Rayleigh, was chosen because of the much higher .