Calculate Terminal Velocity

Ever wonder why skydivers eventually reach a maximum speed when falling, even though the force of gravity in a vacuum will cause an object to continually accelerate? A falling object will reach a constant speed when there is a restraining force, such as drag from the air. The force applied by gravity near to a massive body is mostly constant, but forces like air resistance increase the faster the falling object goes. If allowed to free fall for long enough, a falling object will reach a speed where the force of the drag will become equal the force of gravity, and the two will cancel each other out, causing the object to fall at that same speed until it hits the ground. This is called terminal velocity.

Steps

Solving for Terminal Velocity

  1. Use the terminal velocity formula, v = the square root of ((2*m*g)/(ρ*A*C)). Plug the following values into that formula to solve for v, terminal velocity.[1]
    • m = mass of the falling object
    • g = the acceleration due to gravity. On Earth this is approximately 9.8 meters per second per second.
    • ρ = the density of the fluid the object is falling through.
    • A = the projected area of the object. This means the area of the object if you projected it onto a plane that was perpendicular to the direction the object is moving.
    • C = the drag coefficient. This number depends on the shape of the object. The more streamlined the shape, the lower the coefficient. You can look up some approximate drag coefficients here.

Find the Gravitational Force

  1. Find the mass of the falling object. This should be measured in grams or kilograms, in the metric system.
    • If you are using the imperial system, remember that pounds is not actually a unit of mass, but of force. The unit of mass in the imperial system is the pound-mass (lbm), which under the gravitational force on the surface of the earth would experience a force of 32 pound-force (lbf). For example, if a person weighs 160 pounds on earth, that person is actually feeling 160 lbf, but their mass is 5 lbm.
  2. Know the acceleration due to the gravity of the Earth. Close enough to the earth to encounter air resistance, this acceleration is 9.8 meters per second squared, or 32 feet per second squared.
  3. Calculate the downward pull of gravity. The force with which the falling object is being pulled down equals the object's mass times acceleration due to gravity, or F = MA. This number, multiplied by two, goes in the top of the terminal velocity formula.
    • In the imperial system, this is the lbf of the object, the number that is commonly called weight. It is more properly the mass in lbm times 32 feet per second squared. In the metric system, the force is the mass in grams times 9.8 meters per second squared.

Determine the Drag Force

  1. Get the density of the medium. For an object falling through the Earth's atmosphere, the density is going to change based on the altitude and the temperature of the air. This makes calculating terminal velocity of a falling object especially difficult, as the density of the air will change as the object loses altitude. However, you can look up approximate air densities in textbooks and other references.[2]
    • As a rough guide, the density of air at sea level when the temperature is 15 °C is 1.225 kg/m3.
  2. Estimate the drag coefficient of the object. This number is based on how streamlined the object is. Unfortunately it is a very complex number to compute, and involves making certain scientific assumptions. Do not attempt to calculate drag coefficient yourself without the help of a wind tunnel and some serious aerodynamic math. Instead look up an approximation based on a similarly shaped object.[3]
  3. Calculate the projected area of the object. The last variable you need to know is the sectional area being presented by the object to the medium. Imagine the silhouette of the falling object seeing when looking up from directly beneath it. That shape, projected onto a plane, is the projected area. Again, this is a difficult value to calculate with anything but simple geometric objects.
  4. Figure out the drag force that is opposing the downward pull of gravity. If you know the velocity of the object, but not the drag force, you can use the formula to calculate the drag force. This is (C*ρ*A*(v^2))/2.

Tips

  • Terminal velocity will actually change slightly during the free fall. Gravity goes up slightly as the object gets closer to the center of the earth, but the amount is negligible. Density of the medium will rise as the object gets down deeper into the medium. This is a much more noticeable effect. A skydiver will actually slow down as the fall proceeds because the atmosphere gets increasingly thick as altitude drops.
  • Without an open parachute, a skydiver would strike the ground at about {{safesubst:#invoke:convert|convert}}.

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Sources and Citations

  1. ↑ http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html
  2. ↑ http://wahiduddin.net/calc/density_altitude.htm
  3. ↑ http://exploration.grc.nasa.gov/education/rocket/dragco.html
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