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Terminal Velocity depends on two things- surface area and speed. These are inversely proportionate.

If both these variables affect terminal velocity, why do parachutes slow you down? Initially you had a small surface area but a fast speed- with the parachute you have a larger surface area but lower speed. You have increased one variable but decreased the other. Therefore why do parachutes decrease speed?

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Did you not answer this yourself: "a larger surface area but lower speed"? Terminal velocity and [maximum] speed are much the same thing. –  Henry May 1 '12 at 7:25

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You say:

Terminal Velocity depends on two things- surface area and speed

but I think you're getting slightly mixed up about the terminology. The drag (i.e. air resistance) depends on surface area and speed, but the terminal velocity is the speed and it just depends on the surface area (and air temperature, density, etc , etc that we'll assume is constant). You say;

with the parachute you have a larger surface area but lower speed

and this is quite correct but the speed is the terminal velocity so with the parachute you have a larger surface area but lower terminal velocity.

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$(Drag) = (Coefficient)\times(Area)\times(Speed)^2 $

When $(Drag)=(Weight)$ then

$$ (Speed) = \sqrt{ \frac{ (Weight) } { (Coefficient)\times(Area) } } $$

So the larger the area the less the speed. What is the question again?

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Terminal Velocity depends on two things- surface area and speed. These are inversely proportionate.

Terminal velocity depends on two things: Drag force and gravity. These very much are not inversely proportional to one another. Terminal velocity is reached when the drag force is equal but opposite to gravitational force.

To a bacterium, drag force is proportional to speed. Bacteria live in extremely low Rayleigh number regimes. We humans "live" in high Rayleigh number regimes. In those regimes (falling coins, flying aircraft, re-entering spacecraft, and parachutes), drag force is proportional to the square of velocity.

Note that I've written drag force rather than surface area. Surface area does have a marked influence on drag force. However, different objects with the same surface area can be subject to markedly different drag forces. The shape of the object also comes into play. A nicely shaped wing will offer much less air resistance than will an object such as a parachute even if the wing and parachute have the same surface area. The coefficient of drag distinguishes the wing from the parachute. The drag force on an object moving at some velocity $v$ is given by $F_d = \frac 1 2 \rho v^2 c_d A$, where $\rho$ is the density of air, $v$ is the velocity of the object, $A$ is the cross section area, and $c_d$ is the coefficient of drag.

By design, a good wing has a very low coefficient of drag and a low cross sectional surface area. By design, a good parachute has a very high coefficient of drag and a large cross sectional surface area. It's the combination of the two factors (coefficient of drag and surface area) that make terminal velocity so low for an object descending with a parachute.

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