A space shuttle launch is pretty expensive. Exactly how expensive depends on who you ask, but if you divide the yearly cost of the program by the number of launches you get something in the neighborhood of half a billion dollars. That’s actually pretty trivial by federal budget standards, but it’s still not chump change. It’s a measure of just how difficult it is to get to space. Let’s try to put that number in perspective by running a few numbers describing the cost of energy. Think of it as a Fermi problem.
An orbiting shuttle has potential energy by virtue of its height above the ground. It also has kinetic energy by virtue of its very considerable orbital speed. It’s a one-line calculation to figure our how much once you look up some numbers.
Potential energy is m*g*h, which for typical shuttle takeoff weights and operating altitudes gives something like 2.13 x 1011 J. The acceleration g due to gravity does vary with altitude, but at shuttle altitudes the variation is small enough that we can overlook it for this rough of a calculation.
Kinetic energy is the usual mv2/2. Plugging some more fairly typical figures I come up with a kinetic energy of 3.27 x 1012 J. The kinetic energy thus dominates the total energy, which is by adding the two is about 3.38 x 1012 J.
Here’s the interesting comparison. Here in Texas, electricity costs about ten cents per kilowatt hour. How much would a space shuttle launch worth of energy cost at that rate? Well, there’s 3,600,000 J in a kilowatt hour, so doing the division we see that much energy costs about $90,750.
Obviously this is a grossly unfair comparison. Evewn simple technology like an incandescent light bulb turns only a very small fraction of its electricity into visible light. Launching a shuttle is not every remotely a matter of just turning electricity into altitude and speed. But it does suggest that there is some room for improvement. I think the ideal method would be a space elevator system, where you actually would be getting pretty close to turning electricity into an orbit. Unfortunately that’s a long way away if indeed it ever becomes practical.
Hmm. Probably would have made a good quiz problem, given just the orbital altitude and separately deriving the orbital velocity from that. Unfortunately that chapter was a month ago. Maybe next semester!