Dot Physics


First, how do things get in orbit? Orbit is a motion in which the gravitational force on an object makes it move in a circle around another object. In order to be in a circular orbit, it takes a certain amount of energy.

Let me just start with some calculations. How fast would a satellite need to be going to be in orbit at the same level as the space station (I just picked that because I know it is about 300 km above the surface of the Earth) Here is a simplified diagram of an object in orbit (not drawn to scale):
![fixed orbit](
Here the only force on the object (satellite) is gravity. The gravitational force can be modeled as:
So, what does this force do? One way to think of this is that it makes the satellite accelerate. Don’t think of acceleration as increase in speed, acceleration is a change in velocity. Since the satellite is moving in a circle, it is accelerating (because its direction of velocity is changing). When an object is accelerating only due to moving in a circle, the acceleration is:
![circular acceleration](

Note that r-hat is in the direction of the center of the circle. Newton’s Second law says:

![Newton 2](

Notice that there is only one force (gravity) and that both the force and the acceleration are in the same direction. For simplicity, I will drop the vector notation and just deal with the magnitudes (this is ok since there is no vector addition). This gives:
![gravity and circular](
Now, I want the speed (*v*):
Here G is the gravitational constant, ME is the mass of the Earth, and r is the distance between the satellite and the center of the Earth (why the center – maybe that will be a story for another day). So, the energy needed to obtain this speed would be the kinetic energy:
![kinetic energy 1](
Note that really this is the CHANGE in kinetic energy that I am interested in, but since it starts at rest, it doesn’t matter. UNLESS you start close to the equator, then you already have some energy. I will (for now) neglect this contribution. The reader (you) can add that in as a homework exercise. Or maybe I will put that on the test.

Great, now that I have the velocity I can design my launching catapult – right? No. Not only does the satellite have to have that speed, but it must also but at that height. How much energy does that take? It is the change in gravitational potential energy (that is one way to look at it). Gravitational potential energy is:
![gravity potential](
Why is it negative? How about I just say that it is negative because it’s an attractive force? That should be good enough for now. The change in potential energy will be:
![fixed delta u](
The two “r’s” are the distance from the center of the Earth to the orbit, and the radius of the Earth.

Putting all this together, the energy needed to achieve orbit:
![e total](
![energy total 2](

Ok, so where is this energy going to come from? It is going to come from the catapult. Maybe it is a lever type catapult, maybe a big sling shot. Either way, I can look at this in terms of the work-energy relationship:
![work energy](
If I assume that the force is in the same direction as the displacement for this catapult launching device, then I can write:
Where the delta E is from before. Now what do I do? Well, let me make some estimates of how far this catapult will exert a force on the satellite. After that, I can estimate the acceleration of the satellite.
I am thinking about a HUGE catapult. A catapult to end all wars. One so big that it will be legendary. How about a catapult the size of the Sears Tower in Chicago?
![256px Sears Tower ss](
The Sears Tower is 527 meters to the top of the antenna. What the heck, I will make it twice as tall as that (because I really want this thing to be GRAND). I will go with an even 1000 meters. Now I can determine the average force exerted on the satellite (if it were a sling shot type catapult, the force is not constant):
![avg force](
Really what I want is the acceleration of the satellite:
Using the following values:
I get the average acceleration of:
![calculated accel](
[calculated on google](*6e24%2F1000)*(1%2F6.4e6+-+1%2F6.7e6)&btnG=Search)
I converted this to the common acceleration unit of “g’s” so that it could be compared to manned rockets. I am pretty sure the Apollo Saturn V rockets had the highest acceleration. [Wikipedia]( lists it as 4 g’s. I imagine that an unmanned rocket could have a greater acceleration, but probably not this high (plus this is the average – if it were a sling shot rubber band type, the initial acceleration would be even higher). Clearly this device could not be used for humans.
How long would it have to be for it to be used with humans? Say an average acceleration of 4 g’s? To do this, the catapult would have to launch the capsule (space craft) over a distance of 71 kilometers (44 miles). Funny.
I was going to calculate how many rubber bands you would need to do this, but this post is ginourmous as it is. Maybe that can be another homework assignment.


  1. #1 Michael
    August 28, 2008

    Don’t use a catapult or a slingshot. Use a long launcher rail, about 44 miles long.

  2. #2 Jack
    August 28, 2008

    Obviously it wouldn’t be a mechanical catapult or slingshot, but how about something that is driven my electromagnets like a maglev type launcher with no friction.

  3. #3 rhett
    August 28, 2008

    Actually, it doesn’t matter what exactly gets it moving. It could be an electromagnet, a sling shot, rockets, or a whole bunch of magical elves, that would still be the acceleration.

  4. #4 William Bunker
    August 28, 2008

    Somewhere we approach the concept of a catapult tall enough to drop the satellite into orbit. This scan be accomplished. What of a catapult that is 2 miles tall? 30 miles tall? 62 miles tall?

    And the rail, someone recently tested a circular rail with a track switch leading to a ramp allowing the satellite to be accelerated gradually, like over days, and then launched up by the rail.

  5. #5 Alex Tolley
    August 29, 2008

    Actually skyhooks that catapult objects to orbit are already well studied. Check out “The Hypersonic Skyhook” by Robert Zubrin, in “Islands in teh Sky” ed. Stanley Schmidt and Robert Zubrin ISBN 0471135615

  6. #6 Right Around
    August 29, 2008

    This article discusses a dozen ways to get a satellite to orbit without using a rocket and considers many of them to be possible with today’s technology.

  7. #7 Right Around
    August 29, 2008

    The slingatron is based on the archer’s sling, kind of like a catapult:

  8. #8 adam
    August 29, 2008

    looks like it would work great for satellites

  9. #9 Jack
    August 29, 2008

    I realize that would still be the acceleration, but the author was implying, perhaps jokingly, building a large mechanical catapult. I was suggesting that using a electromagnetic maglev strategy, would be more practical and while it would still be quite large, it would be more feasible to build.

  10. #10 Ventorlin
    October 2, 2008

    There is no magic skill.

  11. #11 Dave
    October 28, 2008

    There is another couple of factors to factor in. One is that, for any such device operating at near sea level, atmospheric friction will eat up a lot of the initial velocity. How high will the initial velocity be to launch into orbit? Somewhere around 17,000 miles per hour (Think of re-entry in reverse!). Think of the heating that something moving at that speed will produce (and, think of the energy loss associated with that heating).

    The other is that you can’t get into orbit by just going up. You have to circularize the orbit. So, once you’ve achieved the altitude, that’s only half the battle, since you then need a horizontal component to remain up there. Rockets achieve this by doing a sustained burn where part of the thrust pushes up, while another part pushes sideways (and, some even do “circularization” burns, too).

    Anyway, orbital mechanics isn’t nearly as simple as it looks like it should be. :-)


  12. #12 rhett
    October 28, 2008


    Good point about the atmospheric friction! I did take into account the energy needed for orbital speed as well as the orbital height.

  13. #13 Inga
    November 5, 2008


  14. #14 Mystery
    February 23, 2009

    Best blog!

  15. #15 romonoeroetoko
    July 8, 2009


  16. #16 romonoeroetoko
    July 15, 2009

    Your news is a cool stuff man, keep it going.

  17. #17 amenodimeno
    July 31, 2009

    That’s good man, keep it going.

  18. #18 queroeropoo
    August 6, 2009

    Good information to me.

  19. #19 adamoerikom
    September 19, 2009

    Stunning blog and good article. High 5 for u man !

  20. #20 Hemal
    March 19, 2010

    Well thanks for the post.. good to know other people are trying being creative… i was also thinking of the maglev type launch..but realistically guys … guess you have all heard of the bloodhound SSC. i soo wish they succeed.

    just imagine the risk involved for attaining such a speed on land and the risk of track defect or sudden climatic change… you would want to maintain that speed for longer than a few minutes… and what about the point of launch where the vehicle would leave the rails?

    i suppose that is why minimising the time of launch minimises risk. might be expensive but the quicker the better i guess

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