Terminal Velocity Calculator

Created by Davide Borchia
Last updated: Jun 23, 2022

Learn why you can drop a squirrel from (almost) any height without causing it much trouble with our terminal velocity calculator.

In this article, you will learn everything you need to know about terminal velocity and even more. Keep reading if you want to know:

  • What is terminal velocity;
  • How to use the terminal velocity formula and how to calculate the final velocity;
  • How to use our terminal velocity calculator; and
  • Why can squirrels fall from the sky with very little damage?

What is the terminal velocity?

Most physics relies on approximations: one of the most commonly used is ignoring the air resistance during the study of a motion, which could potentially lead to the speed of light and high lorentz factors. While for lower speeds, it is totally fine, when a body moves fast enough, it starts experiencing some effects due to drag. You can appreciate this phenomenon even on a bike!

The air resistance assumes particular relevance when we are studying the motion of a body falling through the atmosphere (or any atmosphere, to be fair). In that situation, the body is subjected to two forces only:

In the absence of air resistance, the body would accelerate indefinitely with an acceleration equal to gg, the gravitational acceleration of the planet where the fall is taking place. Let's assume that we are on Earth, where g=9.81 m/s2g =9.81\ \text{m}/{s}^2.

However, in the presence of drag, we witness an entirely different scenario. Since the drag increases with speed, we can identify a point where the net force acting on the body is zero. We reached terminal velocity.

Now you know what terminal velocity is. Let's learn how to calculate the final velocity of a body!

The terminal velocity equation

To calculate a body's terminal velocity, we need to know some parameters, both from the body falling to the ground and the planet toward it is falling.

For the body, you should know:

  • The mass mm;
  • The cross-sectional area AA, that is the surface of the body if you were to look at it from below while assuming it to be "flat"; and
  • Its coefficient of drag CdC_\text{d}, a parameter depending on the body's shape. Flat, angled bodies have higher CdC_\text{d} than round bodies, while streamlined bodies have the lowest CdC_\text{d} possible.

For the planet, all you need to know is:

  • The gravitational acceleration gg (consider the one at sea level, if there's a sea); and
  • The density of the atmosphere ρ\rho: a thick, dense atmosphere (like Titan's one) slows you much more than a thin one.

We can now introduce the equation for the terminal velocity. The first step requires us to consider the formula for the total force acting on a body with mass mm:

F=mg12ρvt2ACd F = m\cdot g - \frac{1}{2}\cdot\rho\cdot v_\text{t}^2\cdot A\cdot C_\text{d}

From this expression, isolate the formula for the terminal velocity:

vt=2mgρACd v_t = \sqrt \frac{2\cdot m\cdot g }{\rho\cdot A\cdot C_{\text{d}}}

How to use our terminal velocity calculator

Our terminal velocity calculator offers you a quick solution to the equation for the terminal velocity.

Insert the parameters of the problem in the opposite field. For the coefficient of drag, you can choose a shape from the presets (sphere, hemisphere, cube, and so on) or insert the value manually.

You can also visit the air pressure at altitude calculator to see how the pressure changes when you're high above the ground.

So, what about squirrels?

Squirrels are fluffy and thin; thus, their area-to-mass ratio is surprisingly high. This implies that the drag on their body during a fall is *relatively) higher than the one of any other animal. The terminal velocity of a squirrel is just 37 km/h37\ \text{km}/{h}, or 23 mph23\ \text{mph}: a squirrel can fall from a tree or jump down a plane, be its own parachute, and land unscathed, while a human would not enjoy the experience due to the huge difference in momentum!

Davide Borchia
Object
Shape
Select...
Mass (m)
lb
Cross-sectional area (A)
Coefficient of drag (Cd)
Environment
Density of fluid (ρ)
lb/cu ft
Acceleration due to gravity (g)
ft/s²
Velocity
Terminal velocity (Vₜ)
ft/s
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