# Free Fall with Air Resistance Calculator

- Not-so-free-falling: plunging in a gravitational field, but with air: free-fall with air resistance
- Calculating free fall with air resistance: formula for the force of friction
- The equation for the air resistance coefficient
- How to calculate air resistance on a falling object: maximum and terminal speed
- Dropping squirrels: application of the free-fall with air resistance formulas

The calculations for free-falling with air resistance are slightly more complex than the ones we use for a vacuum. Follow our short article to learn why. From an introduction to the force of air resistance to the formulas, we will learn how to calculate the air resistance on a falling object, the parameters of the fall, and the possible results. To conclude, we will talk about squirrels.

## Not-so-free-falling: plunging in a gravitational field, but with air: free-fall with air resistance

Physicists like to analyze a problem starting from a very idealized version of the real world. No outside forces, no interactions: studying this situation is much more manageable, and you can add more "realistic" elements.

The same applies to **falling in a gravitational field**. We already analyzed in detail the situation where **air resistance is negligible** in our free fall calculator.

How does the situation change if we include air resistance in our formula to calculate the dynamics of a falling body? Well, from a general point of view, two factors play into a fall with the effects of air resistance (or any other fluid): the **force of friction** and **turbulence**. However, studying fluids is not easy: follow us to discover the free-fall with air resistance equations and explanations.

## Calculating free fall with air resistance: formula for the force of friction

The first thing we can calculate is the **force of air resistance**: its formula is straightforward but with a twist.

Where:

- $F$ — The
**force of air resistance**; - $k$ — The
**air resistance coefficient**; and - $v$ — The
**instantaneous speed**.

As you can see, the force depends on the **square of the speed**: the increase is not linear, but the effect of air resistance is felt more heavily at higher speeds.

## The equation for the air resistance coefficient

The value of the air resistance coefficient is of fundamental importance in the calculations of free fall with air resistance. The value of $k$ contains all the information related to the body and the fluid. Here is its formula:

Where:

- $\rho$ — The
**density of the fluid**where we are calculating the free fall with resistance: air, water, methane, etc.; - $A$ — The
**cross-sectional area**of the falling body; and - $C$ — A dimensionless
**drag coefficient**related to the shape of the body.

🙋 For a deeper analysis of the value of the **drag coefficient**, visit our drag equation calculator.

## How to calculate air resistance on a falling object: maximum and terminal speed

Once you have the result of the force of air resistance formula, you can make a first qualitative observation. If the value is greater or equal to the one of the gravitational force, the body has reached its **terminal velocity**, the maximum velocity allowed in that fluid. We discussed this phenomenon in greater detail at our terminal velocity calculator.

To find the terminal velocity $v_{\text{t}}$, equate the two formulas (for air resistance and gravitational force):

And isolate the value of the speed:

If the body doesn't reach the terminal velocity, we can calculate the **maximum velocity**. To do so, calculate the net force, and then we set up a differential equation. The solution is:

$t$ is the time of fall. We calculate it with the formula:

Where $\text{acosh}(x)$ is the **inverse hyperbolic cosine** function, and $h$ the **height of the fall**.

As you can see, both speeds (terminal and maximum) depend on the **mass of the falling object**. We are finally out of the ideal free-fall regime, where the mass doesn't affect the velocity but only the force, and the behavior modeled by our free-fall with air resistance equations better reflects the one we can observe in real life. Let's drop something now!

## Dropping squirrels: application of the free-fall with air resistance formulas

We like squirrels, and we like how they behave when falling. They are, in fact, one of the only mammals able to survive a fall from any height (we are not sure about bats). How so? They are **fluffy**. Tremendously fluffy. All their fur contributes to increasing the drag force, and they can achieve a non-lethal terminal speed.

Let's find the parameter for a squirrel. For the drag coefficient, we will use the one of a skydiver; it should be fine: $C_{\text{D}} = 0.24\ \text{kg}/\text{m}$. The mass is $m = 0.450\ \text{kg}$, our squirrel is a fit one. The critter got , and lost his balance from a $h = 2.5\ \text{m}$ tree. At which fall will he reach the ground?

Our squirrel will be fine. Now let's climb on an SR-71, a plane able to cruise at $26\ \text{km}$ of altitude. Give a pressure suit to our almost squirrelnaut, and wish him good luck. Jump!

We will reach terminal speed, this time: let's find it.

It's barely more than the speed we found after the jump from the tree! Squirrels are awesome!