Air Resistance and Mass
In our physics class, we learned that air resistance
affects the acceleration of an object, and surface area affects air
resistance. So, if I crumple up a sheet of paper and drop it at the
same time as a flat sheet of paper, the crumpled up sheet will fall
faster. However, if I drop a textbook and a flat sheet of paper with
the same surface area, the textbook will fall much faster. How does
mass affect air resistance?
In fact, mass does not affect air resistance! But it does affect how an
object moves with respect to air resistance. Consider the forces at work
here. Pulling down is the weight of the object. Pushing up is the
aerodynamic drag, which is a function of the surface area, air density, and
speed to the 2nd power (we will assume both the paper and the book are the
same area and do not tumble as they fall, to make things simple). We can also
safely assume buoyancy forces are negligible for these objects. If we drop
the paper, it will quickly reach a terminal velocity where the aerodynamic
drag force pushing up is equal to the weight pulling down. Instead of
falling at a constant acceleration equal to gravity (9.8 m/s^2), it
initially accelerates and then reaches at a constant speed. For the book,
however, the weight pulling down is much greater than the drag pushing up,
at least in the few seconds after you drop it. In this case the aerodynamic
drag is so small compared to the weight that it does not affect the motion
very much and so the book falls at a nearly constant acceleration equal to
gravity, 9.8 m/s^2, until it hits the floor.
So for two objects that have the same aerodynamic drag but different
weights, they will accelerate at different rates through the air until they
each reach their own terminal velocity where drag = weight. And this is also
true of your first experiment where you took two objects of the same mass
and changed their area.
If you were to draw a picture of your experiment, and draw the arrows (or
what is called vectors) of the forces applied to the text book and a flat
sheet of paper, you would have a clearer understanding of how mass affects
air resistance. Let us describe the forces acting on both the falling text
book and flat sheet of paper. The forces are opposing and they are a) the
forces of gravity and b) air resistance.
Observing the force of gravity, F(g), pointing down. For the text book
F(g)= M x g. Whereas, for a flat sheet of paper, the force of gravity
pointing down, F(g) = m x g.
Now consider the force of air resistance, F(a) = Pressure x Area. Since
the text book and the flat sheet of paper have the same surface area, and
the pressure is the same for both, then it follows the force of air
resistance is the same.
Finally, the mass of the textbook, is large enough, such that, the force
due to gravity on the textbook is much, much larger and therefore overcomes
the force due to air resistance. Or even simpler, the textbook has N times
more, flat sheets of paper, and therefore has N x more, the force of gravity.
Just picture two objects approaching on a collision (in this case the force
of gravity and the force due to pressure). The larger one will win, and the
opposing force can either lose instantaneously, or put up a "fight" and lose
Hope that helps.
The book and the paper have about the same air resistance.
The mass does not affect the resistance.
But the book applies a larger force to that resistance,
and so it obtains a higher velocity.
Think of putting your hand in a pool of water and moving it, open palm forward.
It has water-resistance you can feel.
If your arm pushes harder, your hand goes faster.
You experience many speeds and force-strengths, but
It is still considered one value of "resistance".
Resistance is a curve, a plot of force vs. speed.
The number-value of the resistance would be
a proportionality constant in the formula that made that curve.
For pure viscous drag,
The curve is a nice straight line.
The formula would be:
force = resistance x speed
For fluid drag like air resistance or water resistance,
the curve is usually a parabola, a square-law curve.
force = resistance x speed^2
You can do a whole experiment quite consistently using only those formulas.
A further Neat Thing:
The fluid resistance can be almost-calculated from other things in the experiment:
air_resistance = [density_of_air] x [frontal_area_of_object] x [streamlined_shape_factor]
The density of air is in there because
the drag is created by stupidly accelerating all the air in the object's path
to the speed of the object,
after which the moving air slings off to the side of the object
and then gradually slows back down to a stop,
wasting the energy used to accelerate it.
The frontal area of the object tells you how much air that is going to be.
(Multiply frontal area by distance traveled and you have a volume of air.
Multiply volume by density and you have the mass of air.)
The "streamlined_shape_factor" (usually called Cd or Coefficient of Drag)
tells you how un-stupid the object's shape was.
There are smarter ways to get the air around the object
that do not accelerate it as much,
and also use the acceleration of air the front to cancel out the deceleration at the rear.
The smartest shape is the classic long-tear-drop shape of a wing cross-section.
It produces only 5-10% (Cd = 0.05 to 0.10) as much drag as a stupid blunt object.
The paper and the book are pretty standard blunt objects,
they will have Cd = 1.0. (100%).
It is possible to get Cd's a little bigger than 1.0, but not much bigger.
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Update: June 2012