Q & A: Parachute Surface Area

what is the effect of surface area on the rate at which a parachute falls?
- Jenna (age 13)
Collegeville, Pa, US

Jenna -

The answer to your question is a bit more complex than you were probably expecting, but here goes...

Part 1
The force of air resistance on a parachute (figuring the parachute is basically spread out flat against the air) is directly proportional to the parachute's surface area, so long as the parachute is moving fairly quickly through the air. That means that if you double the surface area you double the force of air resistance:

F_{air resistance} = SurfaceArea x OtherStuff

Part 2
Now's where we've gotta do a little math... The net downward force on the parachuter is the force of gravity minus the force of air resistance. So:

F_net = (9.81 m/s^2 x Mass) - (SurfaceArea x OtherStuff)

The net force on the parachuter determines their acceleration (the rate at which they speed up or slow down): F=ma, so...

(Mass x Acceleration) = (9.81m/s^2 x Mass) - (SurfaceArea x OtherStuff)

Do a little rearranging and you get this:

Acceleration = 9.81 m/s^2 - (SurfaceArea)(OtherStuff)/(Mass)

Part 3
Getting from knowing the Acceleration to knowing the actual Speed is tricky, because that "OtherStuff" actually includes the parachuter's velocity. I'm not going to go into the details of solving that for you here, but i think the basic gist should be clear:

The Moral of This Story
More Surface Area = Falling More Slowly

-Tamara

(published on 10/22/2007)

Is the time taken for a parachute to fall proportionally related to the surface area of the parachute? That is, if the surface area is doubled does the time taken to fall double? Why/why not?
- Anonymous

The short answer to your question is that, after a certain point, doubling the size of a parachute does not affect the time of fall (or your velocity).

A really amazing experimental explanation of this is here:

https://www.youtube.com/embed/4CdbBHzw8yQ

The parachute very rapidly acquires almost its terminal velocity, reached when the air friction is enough to just cancel the force of gravity. The friction is proportional to the area. If the parachutes are made out of the same cloth, their masses (and hence the gravitational forces) are also just proportional to the area. If there's an extra weight on the parachute, the fall will slow as you make the parachute bigger until the parachute's own mass is large compared to the other mass.

The time is approximately proportional to the inverse of the terminal velocity, so it's approximately proportional to Area/M_total =  A/(M_p +M_w), where A is the area, M_p is the parachute mass and M_w is the other mass. Once M_p  is much bigger than M_w. this reaches a constant. That means that the velocity for a parachute with twice the area (and twice the mass) is going to be  the same as for the original parachute. If, on the other hand, the parachute were very light compared to the load, then doubling its area would approximately double the time to fall.

Hope that helps!

John +mbw

(published on 11/13/2010)