How Much Fuel Does an Airplane Use?

One of the interesting things about learning about thrust for a rocket, is that we can use the same types of formulas for airplanes. So, let’s take a few minutes and figure out how much fuel an airplane uses to travel from one place to another. We can also look at why airplanes fly at the altitudes that they do, and how the wind affects the fuel used.

How to actually do this?  Well, we need two equations that we have talked about on different posts: (1) the equation for thrust; and (2) the equation for the drag force. When a plane is traveling between two places at a constant altitude, we can ignore the forces in the vertical direction, since the gravity of Earth is balanced by the lift from the wings.  In the horizontal direction, the forces are also balanced (since we are traveling at a constant velocity), namely the thrust of the airplane is balanced by the drag force from the airplane moving through the air.

Screen-Shot-2014-03-25-at-1.45.04-PM
A Boeing 747 for Delta Airlines.

So, what is the drag force on an airplane?  Well, we can calculate it using the formula: F=0.5*Rho*Area*DragCoefficient*Speed², where Rho is the mass density of air, Area is the frontal area of the airplane, DragCoefficient is the Drag Coefficient of the airplane, and Speed is the speed of the airplane (with respect to the wind). Let’s do a simple example, and take the Boeing 747, like the plane shown above. Some assumptions about the 747:

Area = 158.3 m² (that is pretty big!)

Drag Coefficient = 0.05 (that is pretty small!)

Velocity = 562 miles per hour = 250 m/s

I basically found these by looking around on the web.

The surface mass density is 1.23 kg/m³. The density decreases pretty rapidly as you go up in the air. At 30,000 ft, the density is roughly 38% of the surface density (0.467 kg/m³).  At 40,000 ft, the density is about 25% (0.308 kg/m³).

Ok, that was a lot of numbers.  Sorry.  What does this mean?  Well, we could talk about the drag force that the 747 experiences. If we do all of the math, and we assume that the 747 is cruising at 40,000 ft, we get a force of 75,750 N, which is 17,030 lbs. If the airplane were to be flying just 10,000 ft lower, the force would be 25,885 lbs, which is much (50%) larger, showing that the altitude that the airplane flies is pretty important.

Now, let’s calculate how much fuel is used during a 6 hour flight (say New York to London). If we assume that it is all cruise (which is a bad assumption, since a lot of fuel is used to take off), how much fuel does the 747 use?

Well, we have to calculate how much fuel a 747 uses each second at cruise speed and at altitude.  Remember that Thrust = MassFlowRate * ExhaustVelocity.  For a rocket engine, the Exhaust Velocity is really the speed at which the gas comes out of the engine.  For a jet engine, that is not really the case, and it is a bit more complicated.  But, let’s skip over that and just take my word that the “ExhaustVelocity” of a jet engine is about 35,000 m/s. (If the exhaust velocity were really that large, it would be pretty dangerous to be around the backend of an airplane!)

To get the MassFlowRate, we can just divide the thrust by the ExhaustVelocity.  At 40,000 ft altitude, the MassFlowRate would be 2.16 kg/s.  A gallon of jet fuel is about 2.7 kg.  So, a 747 uses just under (80%) a gallon of jet fuel every second.  Depending on your point of view, this is either a lot (a car uses a gallon every few hours), or a tiny bit (a rocket uses a hundreds of gallons each second).

Over the course of a 6-hour (6*3600 seconds) flight, the airplane would  use about 17,300 gallons of fuel (not counting takeoff and landing) if it flew at 40,000 ft.

If the airplane were to fly at 30,000 ft and keep the same exact speed (562 mph), the airplane would use 26,300 gallons!  That is 9,000 gallons of jet fuel more, just for flying at 30,000 ft.

Hopefully this helps you understand why airplanes fly as high as they can.  If you are on a very large airplane that is flying a long way, then the airplane may raise the altitude a couple of times as it uses fuel.  A super heavy 747 can’t fly at 50,000 ft, since its wings can’t support the lift at 50,000 ft.  As the 747 uses fuel and is less mass, it can fly at higher and higher altitudes.  The best track would be to fly at the highest altitude all of the time, increasing altitude all of the time, but rules stop this – there are certain altitude “lanes” that planes can fly in.

Just for fun, if the airplane is at 40,000 ft, it gets about 0.195 miles per gallon. At 30,000 ft, it gets about 0.128 miles per gallon. If the flight had 400 people on the 747, and it flew at 40,000 ft, then each person would get the equivalent of about 78 MPG. Not really that bad! It would be hard to drive somewhere for this type of fuel economy!

Interestingly, if a 747 were to fly at ground level the entire flight, it would use 69,000 gallons of fuel to fly from New York to London, or would get about 0.05 miles per gallon. Yikes!

Finally, how does wind effect the amount of fuel used?  Well, a 747 goes 562 MPH not with respect to the ground, but with respect to the background wind.  So, if the 747 is flying in the jet stream, which can be about west-to-east at 100 MPH, then the ground speed of the 747 flying from New York to London would be 652 MPH, but coming back from London to New York, the ground speed would be 452 MPH.  This doesn’t cause the amount of fuel used per second to change, but it changes the number of seconds the airplane is in the air. From New York to London, the flight would be shortened to 5:10, and back to New York, it would be lengthened to 7:30.  The amount of fuel used would be 14,700 gallons (saving 2,600 gallons, NY to London) or 21,000 gallons (costing about 3,700 more gallons, London to New York).

Ah, physics. I love you.

Oh, on a side note, think about in the first Iron Man movie when he (Iron Man) flew from Los Angles to the Middle East in his suit.  Obviously it must have been pressurized, since he would have to fly at incredibly high altitudes. Iron Man is quite a big smaller than a 747, but he probably was flying about twice as fast as a 747.  So, if you do the calculations, he would have had to use about 147 kg of fuel.  If this was jet fuel (which it was not, but that is a separate discussion) it would be about 55 gallons.  Where did he put all of this fuel??? Marvel Fans want to know!

 

What I do. Simply.

I was asked to give a five minute talk describing what I do at a conference of people who study similar things as I do. Specifically, I was asked to give a talk describing modeling using only the 1000 most common English words used.  They gave two web sites, which don’t match each other exactly, but one is the list and one is a page in which you can enter text and it will tell you which words are “less simple”. (I copied and pasted all of the words from the list into the simplewriter web page and it spit out a ton of them.  So, I made sure that all of my words were not “less simple”.  This is a new fad in science, since it is sort of fun to try to explain what you do using vocabulary that everyone understands.  The problem is that sometimes you have to jump through some very big hoops to say simple things like “model”. The original idea creator of this has a book called Thing Explainer.

Anyways, hopefully you get the idea.  Here is a description of my occupation, only using some of the most common words in the English language:

My Job

I like to explain the hot air in space with a computer: space whether (funny because it is the wrong word!🙂

It is fun, but hard. When the computer does not agree with another approach, it does not make me happy (like sh*t).

One thing to note: the sun does knock off some tiny bit from some air and does give it power. Other air – not so much. Air down low doesn’t have much power. Air up high does. Tiny air bit from space does hit air and does give it power too. Air does move fast. It is hot.

How to explain the hot air and space with a computer?

Break the hot air way up in the sky (or space) into a box and another box and another box and another box and another box…

Then idea is: First consider how fast air does move in one direction, then another, and another. Then consider how much air is fast and slow at the same time (hot or cold). Consider how much air will hit other air and air with a tiny bit taken off. Look at how some air does turn into other air. Then how thick the air is and how much has a tiny bit taken off. Pass to another computer. Repeat. Repeat. Repeat. Like. A Lot. Write out. Draw picture. Another picture. Another.

Do a dance when it does work. Good me.

Give a little cry when it doesn’t. Stupid computer.

Say “My air and space thought computer is the best!”

Show movie of air to work people.

Ask for money. Cry. Because no money.

Thrust

A couple of posts ago, we discussed the idea of Newton’s Third Law: for every action, there is an equal and opposite reaction.  Let’s figure out how to mathematically apply that to getting a rocket off the ground and investigating how airplanes fly!

The main idea with thrust is that if you throw enough stuff out the back at a high enough velocity, it will propel you forward. It is therefore likely that the thrust that is experienced is somehow related to the velocity at which the material is expelled and the mass of the expelled material. This is almost exactly right, except that when we are talking about a rocket, or even an airplane, the stuff that is coming out is a stream of material, and so the mass is not really just a mass, but a mass flow rate (i.e., the amount of stuff per second). In other words, thrust is equal to the mass flow rate times the exhaust velocity, or:

thrust_equation

For a normal, everyday rocket, the exhaust velocity is roughly equal to about 3,000 m/s (meters per second, or about 6,700 MPH).  This exhaust velocity doesn’t vary very much, and it could be as low at 2,000 m/s or as high as about 4,000 m/s, but those are about the limits for a “normal” rocket engine. I will talk about why in further posts, but it mostly has to do with chemistry.

On the other hand, the mass flow rate can vary a large amount from rocket to rocket. That is because the only knob the rocket manufacturers have (if the exhaust velocity is about fixed) is the mass flow rate.  A smaller engine will have a smaller mass flow rate, and there won’t be able to lift large things off the ground.  A large rocket, like the Saturn 5, the largest successfully flown rocket ever, has a huge mass flow rate, and was able to lift a gigantic amount of stuff off the ground (elephants in space?).

My favorite example of a rocket is the V2, which was the first really “useful” liquid propulsion rocket ever made.   This was the rocket build by the Germans in WWII to bomb the Allies.  I will discuss these in more detail in another post also.  But for now, I would like to use this as an example to look at some aspects of thrust and getting a rocket off the ground. So, let’s get the stats on a V2 rocket:

  • With no fuel and no warhead, it had a mass of about 12,500 kg;
  • The warhead was about 1,000 kg;
  • It carried about 8,800 kg of fuel;
  • The mass flow rate about 110 kg/s; and
  • The exhaust velocity of the engine was about 2,400 m/s.
V2
A V2 Rocket. It was extremely advanced from about 1944 to 1957.

With this information, we can calculate some things about the rocket.  For example, the most important thing to calculate is whether the rocket can actually lift itself off the ground. To do that we calculate:

  1. The total mass of the rocket at launch, which is equal to the mass of the rocket, the warhead, and the fuel, which is 22,300 kg.
  2. The weight of the rocket, which is just the mass times the acceleration due to gravity (-9.8 m/s²), which would then be -218,540 N.  It is negative, since gravity is pulling down on it. In order to lift off the ground, the rocket engine has to produce more than 218,540 N of thrust.  That will just overcome gravity, and cause the rocket to lift into the air.
  3. The thrust of the rocket, which is given by the formula above.  We know that the exhaust velocity is 2,400 m/s, and the mass flow rate is 110 kg/s. The thrust is therefore 2400*110 = 264,000 N, which is larger than 218,540, so the rocket will definitely move upwards.

In order to actually calculate that rate of acceleration, we can do a bit more math, using Newton’s second law, which is F = m * a, or force equals mass times acceleration, but we can rearrange it to calculate the acceleration (a = F/m).  The force is the total of the weight of the rocket (-218,540N) plus the thrust of the rocket (264,000N), which is 45,460N.  The total mass of the rocket is 22,300 kg, so the acceleration is 45460/22300 = 2 m/s² upwards.

Now, the cool thing about this has to do with the mass flow rate.  The mass flow rate literately means that the rocket is becoming less massive every second. For the V2 rocket, it is becoming 110 kg less massive every second.  So, after one second, the rocket has a mass of 22,190 kg.  After two seconds, it has a mass of 22,080 kg. And so on, for about 80 seconds, until the rocket has a final mass of 13,500 kg. During the final second of thrust, the acceleration will be quite different than the first second.  In the last second, the rocket weighs about -132,300N, while the thrust is exactly the same as before (264,000N), so the difference is 131,700.  The mass of the empty rocket is 13,500 kg, so that the acceleration just before the thrust cuts off is about 9.8 m/s² upwards, about 5 times larger than the starting acceleration!  This increasing acceleration effect happens with all rockets that have a constant mass flow rate – as they use more fuel up, the accelerate upwards faster and faster.  It is no real surprise then that when rockets are sitting on the launch pad and fire their engines, they look like they are just crawling upwards – because they are!  As the rocket loses more and more fuel, it gains speed rapidly.

mass_flow
An illustration of a rocket shooting mass (80 kg – should be 110 kg for the V2 – oops!) at a given velocity (2200 m/s) every second. In 80s, the rocket is empty, but in that last second, it is accelerating upward fastest.

For manned missions, it is important that the acceleration not become too large, since people can’t withstand huge accelerations.  People tend to pass out when they are accelerated at rates that are about 5 times Earth’s gravity.  And the point of maximum acceleration doesn’t happen at the beginning, but near the end of the thrust.

Ballistic Motion

Ballistic motion is an important concept in our path to understanding how rockets get up into orbit. Really, there are a few types of rockets: (1) rockets that just go up and then come right back down, otherwise known as ballistic missiles; (2) rockets that put something into orbit; and (3) rockets that take something away from the Earth and put it on a trajectory to somewhere else.  Each of these requires more energy than the last one, with the ballistic missile requiring the least amount of energy.

But, what is ballistic motion?  It is the motion that something feels when the “only” force acting on it is gravity. (I say “only” because often atmospheric drag is acting on it also, but we will ignore this for now.)

Let’s take a person throwing a baseball as an example.  Figure 1 illustrates a person getting ready to throw a ball.  (My son Alan drew most of the images again!)

Ball 1
A person getting ready to throw a baseball.

The person then moves their arm forward, accelerating the ball up to some speed.  Typically, this speed is roughly parallel to the ground. Figure 2 illustrates the person’s hand accelerating the ball (wow, that is a beautiful hand!)  When the ball leaves the person’s hand, it is moving with a velocity of Vx parallel to the ground.  In addition, gravity is acting on the ball, so it starts, immediately, to accelerate towards the ground at a rate of 32 feet/sec per second.  The ball will follow an arched trajectory, with the velocity towards the ground growing and growing all of the time, but with the velocity parallel to the ground (Vx) staying the same all of the time.

 

Ball 2
A person throwing a baseball. The person accelerates the ball up to some speed, then lets it go. At that point, it starts falling towards the Earth, but still moves with a speed of Vx parallel to the ground.

Because I am in America, where we are lovers of guns (although I am not), we should use a gun example! Imagine a person shooting a bullet towards a target.  If the person is far enough away from the target, or the bullet is slow, gravity will have enough time to pull the bullet down, and the person could miss the target.  A person far away from a target with a low-muzzle-velocity gun, has to aim upwards to compensate for gravity.

Target
A person shooting a gun directly towards a target will miss the mark, because gravity pulls the bullet down.

A much better example, in my opinion, is a catapult, which has an extremely slow speed, so that all objects need to have a very large upward velocity, in order to actually get the object to where you want it to go.

Cow
A catapult is a perfect example of a machine that relies on ballistic motion to crush enemies. With cows.

Ballistic missiles (or InterContinental Ballistic Missiles, ICBMs) operate on exactly the same principle as the baseball, catapult, or bullet.  Each goes through an acceleration phase, in which something is giving it an initial velocity (the rocket engine, which thrusts for a short amount of time). Then, the force cuts out, and the “only” force left is gravity. Gravity acts to decrease the upwards velocity down to zero, then causes the object to fall at faster and faster speeds. Just like the catapult.

Free Fall 1
A ballistic missile goes through an acceleration phase, then a free fall phase, where gravity is the only force acting on it.
Free Fall 2
A ballistic missile does not actually thrust through the vast majority of its flight!
Free Fall 3
When it lands, the ballistic missile is moving quickly, and typically causes quite a bang.

ICBMs are not the only types of ballistic missiles being developed right now.  There are many companies that want to take “space tourists” on a very fast ride (like 5 minutes).  These companies are creating reusable rockets that have a ballistic trajectory, taking the tourists to about 60 miles into the air, and bringing them back down safe and sound.

Ballistic Motion - Page 8
After the rocket engines turn off, the rocket is traveling under ballistic motion, so it is in free fall, and the people inside will be weightless.  That will continue until the rocket re-enters the atmosphere and the rocket is slowed down by atmospheric drag.  It is at this point, in which the people will experience the most acceleration! (This picture was drawn by me, and not by Alan. Notice the difference in quality. Which is vast.)

The rockets work exactly the same as ICBMs, in that they accelerate for a short amount of time (maybe 100 seconds), and then go into a free fall phase, where the only force acting on the rocket is gravity.

In reality, what happens next is that the rocket, which is well above the breathable part of the atmosphere, keeps going up for a while, reaches its maximum altitude, comes down, and re-enters the atmosphere.  At this point, the rocket is moving at very fast speeds, and starts to feel an incredible drag force.  The people inside the rocket actually have to lay down, since the forces acting on their bodies become so large.  The rocket is slowing down at such a fast rate that the people weigh about 3 times their normal weight.  Gravity is still acting to pull them down towards the ground, but the drag force is rapidly slowing them down.

The space tourists get a large force on them as they take off, and an even larger force on them when they re-enter the atmosphere.  It is truly a wild ride!

Lunar Eclipse

Tomorrow night, which is Sunday, September 27th, 2015, there will be a lunar eclipse that starts around 8 PM Eastern Daylight Time. I am sure that most people know what a lunar eclipse is, but I thought that it might be interesting to discuss the phases of the moon in relationship to the lunar eclipse.  Also, it is a good excuse to talk about the dark side of the moon.

Let’s first discuss the phases of the moon. The illustration below shows four phases of the moon, which are spaced roughly one week apart for a month. The figure is obviously not to scale, but the relative positions of the Earth, the sun and the moon are roughly accurate.  When the moon is furthest away from the sun, so that the order of the bodies from left to right, go moon, Earth, sun, then the moon is completely full.  This is because the sunlit side of the moon is completely facing the Earth, allowing us to see the whole half of the moon.

An illustration of the phases of the moon, showing the moon in four different positions (roughly every week for a month)
An illustration of the phases of the moon, showing the moon in four different positions (roughly every week for a month)

When the moon is between the Earth and the sun we can’t see the moon at all, and it is termed a new moon. Half of the moon is still illuminated by the sun, but we just can’t see that side, since it is facing away from us.

When the moon is off to the side of the Earth, we can see a quarter of the moon, since we see roughly half of the part that the sun is shining on.

Interestingly, you can only see the moon during certain times of the day, depending on the phase.  For example, during a full moon, you can only see the moon when it is dark out (like the person standing on the dark side of the Earth in the illustration – they can see the full moon). That is because, if you are on the dayside of the Earth, then you can’t see the moon, since the Earth is between you and the moon.  If you are at dawn or dusk, the full moon will be on the horizon (opposite to the sun).

The exact opposite is true for a new moon (not that you can actually see the new moon, but when there is a tiny sliver of the moon you can see) – you can only see a new moon during the day – never at night. If you are on the night side of the Earth, the Earth would be between you and the new moon.  If you are at dawn or dusk, then you could see the new moon just on the horizon (close to the sun).

Now, the “dark side of the moon” is a complete misnomer. As you can see from the illustration, the dark side of the moon changes all of the time.  Really, the “dark side of the moon” is simply the side of the moon that we can’t actually see. Ever.

It is theorized that the Moon was once part of the Earth. A long time ago a gigantic asteroid slammed into the Earth, and ended up ejecting a chunk, which ended up being the moon.  As the moon was cooling down and orbiting around the Earth, a heavy part formed. The heavier side of the moon faces the Earth all of the time, so that the moon is considered “phase-locked” with the Earth, meaning that the same side of the moon faces us all of the time. This is illustrated below, with a little circle showing the heavy part of the moon, which constantly faces the Earth.

An illustration of the phases of the moon, with the same side of the moon facing the Earth highlighted.
An illustration of the phases of the moon, with the same side of the moon facing the Earth highlighted.

The side that is facing us is dark sometimes (new moon) and in sunlight sometimes (full moon).  The “dark side of the moon” is simply the side of the moon that we can’t see, since we only see one side of the moon ever.  It is dark about half the time and in sunlight about half the time, just like the side of the moon that is facing us. The first time that humans ever observed the side of the moon that we can not see is when the USSR’s Luna-3 orbited the moon in 1959 and sent pictures back to Earth.

Mercury is phase-locked with the sun, so that the same side of Mercury faces the sun all of the time.  This makes one side of Mercury super-hot, and the other side of Mercury super-cold.  It is theorized that if Earth were phase-locked with the sun, then no life on Earth would have formed, since one side would be extremely hot, and the other side would be extremely cold, making it an inhospitable hell-hole. Since the Earth doesn’t have any really heavy bits to tug one side towards the sun, we lucked out and have a nice 24 hour day. Perfect for humans!

A lunar eclipse is when the moon passes into the shadow of the Earth, as illustrated below.

A lunar eclipse!
A lunar eclipse!

One might imagine that a lunar eclipse would happen every single time that there is a full moon.  It would if the Earth and the moon were always in the same orbital plane, but they are not. The moon’s orbital plane is tipped with respect to the Earth’s orbital plane, so that when there is a full moon, it is almost always either a bit above or below the Earth’s shadow.  There are only a few hours of each month in which the moon is in the orbital plane of the Earth; and it happens that this month those few hours are when it is a full moon, and it will pass into the shadow of the Earth. A relatively rare treat.

Hopefully it won’t be cloudy where you are. We are supposed to be partly cloudy, which seems like the best you can ask for in Michigan.

Newton’s Laws and Making a Rocket Go

In about 1687, Isaac Newton came up with three laws:

  1. An object at rest tends to stay at rest and an object in motion tends to stay in motion unless an external force acts upon it.
  2. A force applied to an object is equivalent to its change in momentum. Most people think of this as being the formula F=ma, where F is the force (a vector, hence it is in bold, meaning that it has both magnitude and a direction, like up or down), m is the object’s mass (a scalar), and a is the object’s acceleration (another vector). Keep in mind that this is a simplification, though! Really, force is a change in momentum, which is the product of the object’s mass times its velocity. This will be discussed below.
  3. For every action, there is an equal and opposite reaction.

The first law is describing something like cars accelerating and decelerating (for example). The second law is sort of a quantification of this – how much force you have to apply to make something accelerate and decelerate. Well, it is mass related. If the object is very big (a aircraft carrier, for example) it takes a LOT of force to accelerate it. If the object is small (a rabbit, for example) it can be accelerated very, very quickly. Newton’s third law is the law that is going to help us get into orbit. The classic example of this law in action is two skaters standing on ice. If they are about the same size, and they push off of each other, they both go backwards with about the same speed.

 

ice_skaters_equal
If two ice skaters who are the same size push off of each other, they will both move backwards with the same velocity. This is because they have the same mass, so both their velocity and momentum are equal.

If you combine the second and third laws, you can envision what would happen if you had a very big skater and a very small skater push off each other – the big skater would not go backwards very fast, while the small skater would move backwards very fast. This is the heart and soul of rocketry!

If the ice skaters are different size from each other, and they push off of each other, their momentum must be equal to each other, so that means that the big person will move much slower than the small person.
If the ice skaters are different size from each other, and they push off of each other, their momentum must be equal to each other, so that means that the big person will move much slower than the small person.

It is sometimes hard to understand the last law in the world we live in. For example, if you push off of a brick wall, the wall doesn’t move backwards. If you push off the ground, the ground doesn’t move. This is really because the brick wall and the world are either too massive for a tiny force to really matter, or the force that you apply are absorbed in a way that you don’t really see. But, take it from me – the forces are really there and they really work. For example, take a very large car and put it on a very flat road in neutral with the parking break off. If you push on it, it will move. Very slowly, but it will move. If you put the parking break on, it won’t move. This is because you don’t have enough strength to overcome the frictional force that is keeping the car in place (neither do I).

Let’s take the example above with the big skater and little skater and generalize it to something that every American can relate too: guns! (Actually, I don’t own a gun, unless you count nail guns, then I own 4 of them!) If you take a big football player and have them fire a tiny gun, then they wouldn’t move backwards hardly at all, but the bullet would move out quite quickly.

If a big person fires a tiny gun, then the kick back is not very much. This is because the momentum of the bullet is small, which means that the football player's momentum would also be small, and since their mass is so large, the backwards velocity would be tiny.
If a big person fires a tiny gun, then the kick back is not very much. This is because the momentum of the bullet is small, which means that the football player’s momentum would also be small, and since their mass is so large, the backwards velocity would be tiny.

Thinking about Newton’s second law, that says that a force has something to do with mass, we can understand this: The bullet has a tiny mass, so it moves forward quickly, while the football player has a huge mass, so he hardly moves at all.

We can describe this in terms of momentum, which is the product of the mass of an object and it’s velocity.  The bullet has a large velocity and small mass, while the football player has a large mass and small velocity.  Newton’s third law basically is saying that when the bullet leaves the gun, it has a certain momentum, therefore, the football player has to have exactly the same momentum in the opposite direction.

This, in essence, describes how a rocket works. The gas coming out of a rocket’s engine has a very small mass (compared to the rocket), but it is moving incredibly fast.  The rocket itself is HUGE, so it moves very slowly (at first).

After a minute or so, the rocket has expelled so much fuel, that the mass has decreased significantly, so that the mass of the rocket and the mass of stuff that is coming out are closer to equal, like this example:

If a little kid fires a bazooka, the kickback would be incredible, since the bullet is big and moving fast, making the momentum quite large. The kid's momentum must be large also, and since their mass is small, the velocity must be quite large.
If a little kid fires a bazooka, the kickback would be incredible, since the bullet is big and moving fast, making the momentum quite large. The kid’s momentum must be large also, and since their mass is small, the velocity must be quite large.

In this case, the mass of the little girl is still much larger than the bazooka bullet, but the bullet is still moving very quickly, and therefore the girl moves backwards pretty quickly also.  The momentum of the girl and the momentum of the bullet have to be equal and opposite.

Next time, we will take a closer look at Newton’s Second Law and learn how to calculate thrust!

(Artwork done by Alan Ridley)

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.)