Escape Velocity
Published on Mar 3, 2008 at 5:41 pm.
1 Comment.
Filed under physics.
Every now and then, you hear physicists, astronomers, or rocket scientists talking about “escape velocity.” So, what is escape velocity?
To put it very simply, escape velocity is the speed needed to pull away from an object’s gravitational reach. Now, let me explain this. You can imagine throwing an object upwards. From experience, everyone knows that what goes up must come down. Whatever object that you throw up will eventually fall back down. 
If you throw it up harder, then it will go higher, but it will still fall back to the ground. However, the farther an object is from the surface of the Earth (or any other body), then the weaker that the gravitational forces will be on it. So, if the object is thrown upwards very hard, then it will go very high, even higher than it would if gravity did not lessen with altitude. That means that it would also take longer to fall back to the ground. If an object is thrown upwards fast enough, though, then it will go so high that the force of gravity will become so little that gravity will be unable to ever make the object fall back to the ground. The minimum speed necessary for that to happen is called the escape velocity. The diagram here shows these cases. Case A shows an object simply thrown upward. Case B shows an object thrown upward much faster. And Case C shows an object thrown upward at a high enough speed to escape the force of gravity and never fall back to the ground.
It should be noted that escape velocity and orbital velocity are different concepts. Also, escape velocity is the speed needed to pull away from the gravity of a body. When in orbit, you are still in a planet’s gravitational field. In fact, it is that gravity that keeps things in orbit rather than allowing them to fly off into space. Escape velocity is much higher than orbital velocity. Also, escape velocity should not be confused with the speed needed to escape Earth’s atmosphere. You can go fast enough to shoot high above the atmosphere, but still not be going fast enough to get into orbit, and nowhere near fast enough to keep going away from the planet. Such a trajectory will be a suborbital one. It will get out of the atmosphere, but then fall right back. That is what America’s first astronaut, Alan Shepard, did in the first manned Mercury spaceflight aboard Freedom 7. It is a common mistake to say that such speeds are in excess of escape velocity because they take the spacecraft out of Earth’s atmosphere. The actual escape velocity is the speed needed to completely escape the gravitational pull of a body. It should also be noted that the escape velocity is a theoretical concept. In reality, an object moving at Earth’s escape velocity from its surface would not likely make it away from Earth. Rather, the atmosphere would be in the way, and the object would likely burn up on the way out of the atmosphere. That is one reason to use rockets to launch rather than simply shooting something from the surface of the planet with a huge gun.
So, is there any way to compute the escape velocity of a body? Yes, there is. I cover that in my introductory physics classes. I won’t go into all the details here, but I’ll at least hit the high points for anyone who might be interested. It all comes down to Newton’s Universal Law of Gravitation. This law says that the force of gravity between two bodies is equal to the product of their masses divided by the distance between them squared and multiplied by what we call the universal gravitational constant (G).
The universal gravitational constant is a term that is determined by our choice of units and by the strength of the gravitational force. If we use the normal SI units (kilograms for mass, meters for distance, and Newtons for force), then G = 6.67×10-11 N m2/kg2. Thus, we can see that the force of gravity decreases as the distance r increases. But, you will notice that this equation never really goes to zero. It approaches zero as r approaches infinity, but since r can never actually be infinity in a universe with a finite event horizon, then it never actually reaches zero. However, it can get pretty small. And, once it is small enough, other forces from other bodies dominate. But, in our derivation of escape velocity in the introductory physics class, we often ignore the effects of other bodies (just to make the calculations easier).
The next thing to consider is energy. The gravitational potential energy is related to how far apart two bodies are. A lot of people know this concept from beginning physics class as the equation U = mgh, where U stands for the potential energy, m is the mass of the body, g is the acceleration due to gravity, and h is the height of the object. But, that is only an approximation that holds in the laboratory. It assumes that the force of gravity (and thus the acceleration due to gravity) is constant. As we saw above, that is not the case. But, over short distances, such as in common experience, mgh is just fine for potential energy. But, when we are talking about escape velocities, we are obviously talking about long distances, so we need another equation. That can be derived using integral calculus, which I do when I teach the majors class. What you get is another equation for potential energy, U, seen here. This time, I’ve also changed notation. Capital M is the mass for the planet (or other large body that we are trying to escape). Lower case m is the mass of the smaller body that is escaping. R is the distance from the center of the large mass (assuming it is spherical). 
Notice that the potential energy goes as 1/R, not 1/R2 as does the force. (Note: for purists, the actual gravitational potential energy should be negative, but I am looking at the change in potential going to infinite distance, and that is a positive value). But, the key is to also look at kinetic energy, K, not just the potential energy. The kinetic energy is the energy of motion, and it is equal to one half of the mass time velocity squared. Anyone who has had a physics class (and a lot of other people, too) know that energy is conserved. So, as an object is thrown upwards, its kinetic energy decreases and its potential energy increases. The value of U that I give here is the maximum value by which the potential energy can increase. So, if the kinetic energy is less than that value, then the kinetic energy will decrease to zero. The object will stop. And then, the object will begin to fall, converting the potential energy once again into kinetic energy. If it were not for the effect of atmospheric drag, an object would fall and return to its original height moving just as quickly as it was when it was thrown upwards. If the kinetic energy is larger than this maximum potential energy difference, then the object will not stop. The kinetic energy will drop, and the potential energy will rise, until the potential energy has reached its maximum value (technically at a distance of infinity). But there will still be kinetic energy left over, and the object will continue to move away from the larger body. 
So, the critical case is when the two values are equal to one another. In that case, the kinetic energy equals the maximum possible potential energy difference. The object will slow to a stop, but it will take an infinitely long time to do so, and it will be infinitely far away at the time. Any slower, and gravity will eventually pull things back together. Any faster, and the object moving away from the larger body will never stop. This speed at which this critical case is reached is called the escape velocity. Notice that the small m cancels in the equation, so it does not matter what mass the smaller object is. So, you just have to solve this equation for v, the escape velocity. What you get is the following equation:
This is how fast you would have to go in order to pull away from a larger body of mass M, assuming that you start at a distance R from the center of the body. Note that while escape velocity is often given in tables as the speed needed to escape from the surface of a body, this equation also works if an object is in orbit. All you need to know is how far the objects starts from the center of the body. Since R is the distance from the center of the body, a lower escape speed is needed to escape from orbit than from a planetary surface. Of course, if you are starting from the surface of a spherical planet, moon or asteroid, then R will just be the radius of the body. But, physics students should have noted that the kinetic energy equation does not have direction. The v in that equation is really speed, not velocity (In physics and engineering, we define velocity as speed in a particular direction.). So, while at the top of this posting I motivated the idea of escape velocity as the speed that you need to throw something straight up, it is not really necessary for it to be moving in any particular direction at all. So, while the term is escape velocity, it really should be escape speed.
To think about this, consider a large gun firing a projectile to the side from a very high altitude on an airless world. In the digram here, you see that the yellow dotted line falls to the ground. If the projectile is fired faster, then it naturally goes farther. If the projectile is fired quite fast, the distance that it goes is farther because the surface of the planet curves away from the trajectory. The object is falling, but the direction of fall is towards the center of the planet, not towards the bottom of the picture. So, if fired sufficiently fast, the projectile may go one quarter of the way around the world before it hits the surface of the world. But, if it is fired even faster, then it might make it all the way around without hitting the surface of the world. It will fall, getting closer to the planet’s surface, but the surface keeps curving away from the path. The projectile “misses” the planet. But, as it fell, it was increasing speed. So, when it “misses” it swings around and goes higher. In fact, the potential energy from falling closer to the planet is converted into kinetic energy, so it ends up going fastest at the far point in its path, when it is closest to the surface of the world. But, then it goes higher, converting the kinetic energy back to potential energy. Eventually, it reaches its starting point, moving the same speed that it started with. It then repeats its motion. This is an orbit. It is not a circular orbit, but it is an orbit.
If the projectile is fired even faster, then it goes into a circular orbit. At that point, the potential and kinetic energies are equal in magnitude (technically one is positive and one is negative by definition, so the total energy is zero). If the projectile is fired even faster, then it is going too fast for that circular orbit and it slings outward from the planet. But, as it goes higher, it slows until it begins to fall again. This is an orbit that is the reverse of the orbit in the diagram above. But, if the projectile is fired fast enough, then it will sling out and never come back. The minimum speed for that to occur is the escape velocity. Mathematically it is the same as the escape velocity for something fired straight up. The direction doesn’t matter. (Of course, shooting it right at the ground wouldn’t really work!)
This is a little more mathematical posting that most of the ones that I’ve done, but I thought it might be interesting to talk about, particularly since the term has been in the news some of late. I heard one reporter using the term escape velocity when referring to the rocket used to shoot down the satellite a while back, even though escape velocity was not involved. The missile only went fast enough to escape the atmosphere (just barely), not fast enough to go into orbit, much less to break away from Earth’s gravity.
-Astroprof
Roy Lee Epp on March 4, 2008 at 8:32 pm: 1
A very enjoyable read. I didn’t know that you can escape in any direction but that makes sense.
Please write more of these columns.
And I especially liked the level of math included.