Calculating the Performance of a Parachute

In model rocketry, a key part of the project is choosing a parachute to safely recovery your rocket after the motor burns out. The usual question is "How big a parachute do I need?" But you might also ask, "I have this parachute already. Will it work?" Finally, you might want to validate a manufacturer's claim of performance, usually given as the drag coefficient of the chute.

All three of these questions can be addressed with one equation using a little algebra.

diameter = sqrt ( (8 * mass * gravity) / (pi * air density * coefficient of drag * descent rate^2) )

The key here is to work in metric! You can convert to English units of feet, inches, pounds, etc. if desired, but it requires some extra work. But, hey, if my high school sophomores can do those conversions, so can you!

- diameter = parachute constructed diameter (meters)

- mass = mass of rocket, spent motor, and parachute combined (kilograms)

- gravity = 9.8 meters / seconds^2

- air density = 1.22 kilograms / meters^3

- descent rate = how fast the rocket is going to come down (meters / second)

- pi = 3.14159

- coefficient of drag = measure of the resistance produced by the parachute (unitless)

Step 1: How Big a Parachute Do I Need?

Let's look at parachute diameter first.

diameter = sqrt ( (8 * mass * gravity) / (pi * air density * coefficient of drag * descent rate^2) )

As written, the equation above solves for how big a parachute you need to bring your rocket down at a specific descent rate given the mass of the rocket. If you have a flat parachute (technically a "parasheet", photo 1), use 0.75 as your coefficient of drag. If you have a domed canopy ("elliptical", "hemispherical", etc., photos 2 and 3), use 1.5 for your coefficient of drag.

Suppose you have a rocket that will have a mass of 1 kg at recovery, an elliptical parachute, and want the rocket to come down no faster than 6 m/s to prevent damage on landing. Plugging those numbers into our equation yields a diameter of 0.62 m (24.3 inches).

Personally, I prefer slower touchdowns, so I'm aiming for more like 4 m/s. To achieve this, I need an elliptical parachute with a diameter of 0.92 m (36.4 in).

Step 2: How Fast Will My Rocket Descend Under This Parachute?

Ok, so it's great that you know how big a chute you should use. But what if you don't have that exact chute on hand but do have another size? You can use the same equation, just rearranged to isolate the descent rate.

We had

diameter = sqrt ( (8 * mass * gravity) / (pi * air density * coefficient of drag * descent rate^2) )

Rearranging to get the descent rate, we now have

descent rate = sqrt ( (8 * mass * gravity) / (pi * air density * coefficient of drag * diameter^2) )

Again assuming our 1 kg rocket, suppose I look in my range box and find a 48" flat parasheet? Remember that a parasheet has a coefficient of drag = 0.75. Plugging this into our equation yields a descent rate of 4.28 m/s (14.1 ft/s), comfortably inside our 6 m/s limit. But suppose I only had a 36" parasheet? In that case, my rocket will come down at 5.71 m/s (18.7 ft/s). While I'm still inside my nominal limit, I would only use this if I had a nice, lush grass field to recover on, not the hard clay ground of my normal flying field in Colorado.

Step 3: Validating the Manufacturer's Drag Claims

A common selling point for model rocket parachutes is the coefficient of drag for any given design. The higher the coefficient, the more drag for any given size of chute. If you have a particularly narrow rocket or are trying to achieve extreme altitudes, you want the smallest chute you can safely use. Alternatively, you might just want to see if the maker is honestly advertising their wares.

In this case, we take our equation

diameter = sqrt ( (8 * mass * gravity) / (pi * air density * coefficient of drag * descent rate^2) )

and rearrange to isolate the coefficient of drag.

coefficient of drag = (8 * mass * gravity) / (descent rate^2 * pi * air density * diameter^2)

Now, we need to have actual flight data to put into the equation to find the actual coefficient of drag our parachute produced during the flight. The best way to get this is to fly an altimeter on board the rocket to collect data during the flight. The altimeter needs to record altitude vs. time to allow you to find the descent velocity. Alternatively, you can perform drop tests with your parachute from a known height and measure time to touchdown. This technique does have some error as it takes a bit of time for the system to achieve a stable descent rate once released.

I manufacture elliptical parachutes for model rockets and advertise them as having a nominal coefficient of drag of 1.5 based on published data for that style of parachute. I recently flew a 1.83 m (6 ft) parachute (photos 1 and 2) on a 4.7 kg rocket that recovered at 4.37 m/s (altimetry data in photos 3 and 4). Plugging this data into the equation above yielded a drag coefficient of, not surprisingly, 1.5, nicely validating the designed performance.

Step 4: Conclusion

Whether you want to determine the size of a parachute, the descent rate it will produce, or validate a manufacturer's claimed drag coefficient, all it takes is one equation and a little algebra.

While I've used model rocketry as the application here, the same principles apply to skydiving, recovering instruments dropped by a high altitude balloon, etc.

Teachers, you might use these equations in a class to investigate different aspects of parachute recovery. For instance, does the length of the suspension lines affect the coefficient of drag? Does the rate of descent affect the coefficient (i.e., for the same chute with heavier or lighter masses)? What other cool investigations can your students perform?