In much of elementary physics, you are told to ignore air resistance. It is complicated to determine; more importantly, it is crucial to first understand motion without air resistance. One of Galileo’s great achievements was that he saw through the confusion introduced by air resistance and perceived that the acceleration caused by gravity is constant for all objects. However, air resistance does exist, and it does affect motion.

The parachutist shown in Concept 1 on the right relies on air resistance acting on the parachute to slow his rate of descent. As the parachutist falls, his downward speed increases until the magnitude of the force of air resistance exactly equals that of gravity. At that point, there is no net force acting on him, and he descends at a constant speed, called his terminal velocity.

In this section, an equation for air resistance is supplied so you can begin to understand its impact on motion. We use a spreadsheet to calculate the effect of air resistance on a falling object. This provides an introduction to motion with non-constant acceleration.

Since we are focused on modeling motion, not the theory of air resistance, we will quickly cover a few essential points on how air resistance affects motion. Air resistance is a force, like the force of gravity pulling you down, or the force you exert when you push a door open. In ordinary conditions on Earth, falling objects experience not only the force of gravity, but also this opposing force. The force of air resistance, called drag, acts in a direction opposite to the object’s motion. (Note: When an object is rising, air resistance, which always opposes motion, acts in the same direction as gravity, and both act to reduce the object’s speed.)

The acceleration due to air resistance “opposes” a falling object’s free-fall acceleration and reduces the overall (net) acceleration of the falling object. Unlike free-fall acceleration, the acceleration due to air resistance is not constant. We use a typical equation here for the acceleration due to air resistance. This acceleration is the product of a drag constant k and the square of the object's speed. The drag constant depends on the object's shape and surface area. Parachutes are designed to have large drag constants.

In Equation 1 to the right, you see the same equation in two formats for determining the speed of a falling object. The notation of the first may be a little unfamiliar but it is based on the first of the four standard motion equations shown in a previous section. The second formulation of the equation is the form we use in the spreadsheet that we discuss below.

The equation comes from rearranging the definition of acceleration. It states that the current speed v_c equals the prior speed v_p plus the product of the net acceleration and the elapsed time. (We use the terms “prior” and "current" here instead of “initial” and "final," for reasons we will explain in a moment.)

In this case, the net acceleration is the acceleration caused by gravity minus the acceleration in the opposite direction caused by air resistance. The overall equation states that the current speed equals the prior speed plus the change in speed due to the net acceleration.

This is not a constant acceleration. The acceleration due to air resistance is a function of the square of the speed, and the speed changes until terminal velocity is reached. The prior speed value is used to calculate the “current” acceleration due to air resistance, which means that for this equation to provide a good approximation, the time increment must be small. This is why we use “current” and “prior” for the speeds: to indicate that they represent the speed in a pair of closely succeeding instants, rather than overall initial and final values.

This is not an easily solved equation (unless you know the right mathematics, namely differential equations). But your computer will let you model the motion without having to solve the differential equation. We used a tool called a spreadsheet to solve the problem. Some well-known spreadsheet programs include Microsoft® Excel, Lotus® 1-2-3™, and the spreadsheets that are part of products like AppleWorks®. You see a portion of the spreadsheet in the illustration on the right.

The spreadsheet solves the problem by dividing the time into small increments. Every 0.01 seconds, the speed and acceleration are recalculated using the formula in Equation 1. Because the time interval is small, this approximates calculating the instantaneous speed and acceleration. It takes 500 iterations to reach 5.0 seconds, but the result is quite accurate.

Depending on your browser, you may be able to launch a spreadsheet program by clicking on the link below. You will then see a spreadsheet calculating the speed at successive instants using the equation described above.

If you open the spreadsheet and click in the middle of the column labeled “Velocity with drag” you will find exactly the second version of the equation shown in Equation 1. We used the spreadsheet’s cell naming feature to simplify its expression, instead of the default spreadsheet equations like “=B1−C2.”

In the spreadsheet, we set some initial values: the value of g, the initial velocity of the object (0.000 m/s), the increment of time we will use (0.010 s) and the value for k (0.200 m⁻¹). You can vary these. (Note that in the spreadsheet we assign “down” to be the positive direction, for convenience in reading the spreadsheet.)

Let’s walk through the first two iterations. After 0.01 seconds, the object’s velocity equals its initial velocity (0) plus the product of g and Δt. This equals 0.098 meters per second. The spreadsheet calculates the effect of air resistance as k times the square of prior velocity. Since the prior velocity equals 0, our model says there is no air resistance in the initial 0.01 seconds. This is an approximation in our model since there will be some air resistance, but by choosing a small increment of time we minimize the effect of this approximation. The spreadsheet tells us that after 0.01 seconds, the object moves at 0.098 meters per second.

Now the spreadsheet iterates again. The prior velocity is now 0.098 m/s. This time, air resistance will be a factor. The spreadsheet subtracts the product of k (0.2) and the square of the prior velocity (0.098) from g and multiplies the resulting net acceleration by the increment in time (0.01). The spreadsheet tells us that the velocity after 0.02 seconds is 0.1959808 m/s, a little less than it would be if we had disregarded air resistance. This is displayed as 0.196 by the spreadsheet, so you will not see any difference between the velocity with drag and the velocity without drag yet. But after 0.05 seconds, you can see the velocities with and without air resistance starting to differ.

We could continue, and the spreadsheet does, hundreds of times, thousands if you wish. If you look further down in the spreadsheet, you see that at 3.62 seconds the object reaches a terminal velocity of 7.000 m/s. Not only does this spreadsheet correctly calculate the velocity at any instant in time, it lets you find a value for the terminal velocity when it occurs.

Click here to open the spreadsheet. If the file does not open, on Windows click with the right mouse button and choose the save option. On the Macintosh, hold down the “control” key and click on the link, then choose the option to download the file.

The great thing about spreadsheets is that they do “what if” analysis very well. You may be thinking that 3.62 seconds seems to be a short time to reach terminal velocity, and also wondering what would happen if you used a smaller value for k. You could use the spreadsheet to calculate: if an object reaches a terminal velocity of 50 m/s, how long will it fall before doing so? Just change k until you see that value become the terminal velocity, and observe the elapsed time.

Once you become familiar with spreadsheets, you can build models like this quite quickly. An experienced spreadsheet user could create the model we used here in less than 15 minutes.