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Terminal Velocity and Drag

One of my favorite forces to think about is air resistance, or the drag force.  I like this force since it is very easy to relate to and it strongly affects the vast majority of people in ways that they probably don’t even realize. (Actually, as a cyclist, I hate this force most of the time, and very much like it whenever there is a tailwind…) And, it is critically important for rockets and space flight, too!

You probably have had some experience with wind resistance.  When the wind is blowing hard, you can feel the force of it.  When you are riding a bicycle, you can feel its effect.  When you are driving a car, you are affected by wind resistance.

Typically, we can think of wind resistance, or drag, as being a force that is applied to an object that is trying to move through a medium.  For example, if you are riding a bicycle, you feel drag from moving through the atmosphere. As you go faster and faster, you have to exert more and more energy to keep going. If there is a head wind (i.e., the wind is blowing opposite to the direction that you are moving, or into your face), you have to exert even more energy.  If there is a tail wind (i.e., the wind is blowing with you, or on your backside), then you exert less energy to go the same speed.

Mathematically, we can write the drag force as (sorry – math!):

F=-½ρΑCV²

ρ (rho) is the density of the medium that the object is traveling through (i.e., air). If this increases, then the force become harder to move through. For example, it is almost impossible to ride your bicycle through water, since the density of water is about 1000 times the density of air.  So, it is literally 1000 times harder to ride your bike in water, even with scuba gear on. Submarines have a top speed of about 30 MPH, while commercial jets have a top speed of about 600 MPH. It is because of the density of the medium they are traveling through.

A is the frontal area of the object that is moving through the medium.  The area is why it is much easier to ride a bicycle at fast speeds bent over, rather than sitting in an upright position. It is also why bigger cars tend to get worse gas milage than smaller cars, or adding things like racks on the top of your car will reduce your fuel efficiency.  Read this page for more car information!

C is the “drag coefficient”, which is sort of difficult to actually measure. It describes how easy it is for the air to flow around the object.  So, something that is more aerodynamic in shape (like an arrow or a wedge) has a much smaller drag coefficient than something that is flat (like a semi-truck) or shaped like a parachute.  You can also change the drag coefficient of an object by coating it with something special.  For example, modern cars are painted with special paints that help the air move more smoothly over the surface. Swimmers now wear body suits that allow the water to move across their bodies easier than their bare skin.  Fish are super slippery to allow them to have a very low drag coefficient in water.

V is the velocity of the object moving through the medium with respect to the medium. It is squared, which means that if you double the object’s speed, the force goes up by four times.  This is why a head wind and a tail wind make such a huge difference when biking. Even a small change in the relative velocity between your speed and the atmospheric speed will have a big difference in the amount of force you have to apply to move at your speed. It is also why flying from the west coast of the US to the east coast takes significantly less time than traveling from the east coast to the west coast – the jet stream is typically from west to east, so you have a tail wind in one direction and a head wind in the other.

One of my favorite subjects to discuss with respect to the drag force is terminal velocity, which is when the force of gravity balances the drag force.  The most common example of this is someone jumping out of an airplane. When a person jumps out of an airplane, gravity starts to pull them down towards the ground, and they accelerate.

Meet Dave. Dave is sort of dumb. He jumped out of an airplane with no parachute. He will quickly accelerate downward, gaining speed. Then the drag force will start to slow the downward acceleration and eventually Dave will reach terminal velocity.

Then, as the person gains speed, the drag force acts to reduce their acceleration (but they are still gaining speed!) As the person falls faster and faster, the drag force becomes larger and larger (because of the V term).  Eventually, at a certain speed, the drag force exactly equals (but is opposite to) the force of gravity.  This is terminal velocity.

terminal_velocity_velocity
Velocity as a function of time for different “objects” falling from a plane. The “Spread Eagle” person quickly reaches a relatively slow terminal velocity, while the person with “Toes Down” takes significantly longer to reach a much faster terminal velocity. With no drag force, a person would never reach terminal velocity.

I know that equations suck, but looking at the equation for terminal velocity gives some pretty interesting insights into some cool things. So, plug your nose and take a look:

Eqn_terminal_velocity

In this equation, V is the terminal velocity, m is the mass of the object that is falling, g is gravity (we can think of this as just 10 for now), ρ is the density of the air, A is the area, and C is the drag coefficient.  So, what does this mean?

Well, if something is heavier (or more massive), it falls with a faster speed. If the object has a larger area, it falls slower.  If it is shaped like a parachute instead of a bullet (i.e., C is larger), it falls slower.  Finally, if the density of the air is larger (like water instead of air), the object falls slower (a bowling ball will fall about 30 times slower in water than in air.)

As the plot above shows, and the equation describes, if Dave falls with his toes down, he will fall faster than if he is spread eagle on his back. This is mainly because his area is increased being spread eagle.

If Dave spreads his arms and legs, he will fall slower than is he points his toes towards the ground. Both his area and his drag coefficient change in this example.
If Dave spreads his arms and legs, he will fall slower than if he points his toes towards the ground. Both his area and his drag coefficient change (larger in the spread eagle position) in this example.

Since Dave doesn’t have a parachute in this example, he will continue falling at a rate that is not conducive to survival. On the other hand, if he were to open a parachute, his area would increase dramatically and he would slow way down.

Stan has a parachute. He will survive the day. Dave, on the other hand, will harshly encounter the ground.
Stan has a parachute. He will survive the day. Dave, on the other hand, will harshly encounter the ground.

If you have heard of Galileo and the thought experiment of dropping a feather and a bowling ball off the top of a tower, you will know that physicists will say that they should hit the ground at the same time. That is only true if there is no air!  If there is air, they behave exactly as you expect: the feather “floats” to the ground, while the bowling ball plummets to the ground. If you want to see how the feather and bowling ball fall in air and in a vacuum, take a look at this video, which is awesome. Seriously, it is so cool.

There are some cool things about this equation:

  • Cats are very light compared to their surface area.  Therefore, they reach terminal velocity very quickly, so can right themselves and land on their feet. Dogs, not so much. They are quite heavy with respect to their surface area, and have a much faster terminal velocity. (While you may think of an experiment to do here, I do not condone what you are thinking.)
  • A cloud, or fog, is just a bunch of water drops. Rain is just a bunch of water drops.  Why does a cloud stay suspended in the air, while rain drops fall?  It all has to do with the mass divided by area in the terminal velocity equation.  For very small drops, the terminal velocity is super small, and the drops are “suspended” in the air, falling at speeds of inches per hour.  As the drops hit other drops and become bigger and bigger, the mass becomes larger faster than the area.  So, the terminal velocity increases. As the velocity increases, the drop hits more drops, absorbing them and becoming even larger, and increasing the terminal velocity more.  Eventually, they are large enough that they fall from the sky as rain.
Large raindrops fall fast, while tiny raindrops fall slow. Super small raindrops fall so slow that they don't seem to be moving.
Large raindrops fall fast, while tiny raindrops fall slow. Super small raindrops fall so slow that they don’t seem to be moving.

How does this all relate to rockets? Well, in the next post, I will talk about that!

(Artwork done by Alan Ridley.)

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