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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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 .
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+