This never came up on a quiz, although I did once take a course on Interplanetary Navigation. See the diagram.
Your spaceship is in an elliptical orbit around the evil planet GOP. In one month’s time the planet is going to explode. You need to get away. Fortunately you have just enough fuel left to achieve escape velocity. You wait for the optimum point in your orbit to fire your thrusters.
What is the optimum point?
Question 2: In which direction do you fire your thrusters?
Question 3: Why?
I’m going to review the solution along with comments that have been submitted to the post and also on Facebook. Two comments were posted to the blog. Here they are:
Here are the comments from Facebook:
Steven Breed If the question is only whether the spaceship escapes or not, then only energy matters because that is what determines whether a binary system is gravitationally bound. If you want to optimize the spaceship’s escape in some way, then its position in the orbit probably does matter.Steven Breed I re-read the question. It’s asking from where can the escape be accomplished using the minimum of fuel (since the spaceship only has the minimum necessary). Expending the fuel at the bottom of the orbit optimizes the result, I believe. The spaceship gains the most at this point because it isn’t burdened by having to carry the fuel back out to higher potentials. I’ve cheated a bit because, although I haven’t read the appropriate Heinlein, I did read this in 2010: Odyssey Two by Clarke some years back.Prasad Golla Yes, that’s right the total energy is the same irrespective of the position in the orbit.
Of course, Steve Breed, with some history of studying physics, nailed it on the second try. A number of things are correct:
- The energy of the spaceship in orbit is constant. While a satellite is orbiting a planet there is nothing adding or subtracting energy (almost). When the satellite is farther from the planet it has higher potential energy with respect to the planet’s gravitational field, but its velocity is lower, and so on.
- A rocket engine is not a constant power device. When a spaceship is still sitting on the pad right after firing its engines, the engines are not developing any power, because there is no forward motion of the rocket. The power delivered by a rocket engine is the product of its constant thrust and its variable forward velocity. Actually, rocket engines develop more thrust when operating outside the atmosphere.
- When the spaceship is at perigee its speed is greatest. If the rocket engines are fired at this point while the spaceship is pointed in the direction of motion, then the rocket engines develop the most power, and the most energy is imparted to the spaceship.
- The spaceship is best able to escape the evil planet GOP by firing its rockets at perigee, with the thrust along the line of forward motion.
This was explained in Robert Heinlein’s novel The Rolling Stones, in 1952. I read it when it was serialized in Boy’s Life magazine, so I was aware of this bit of science before I took a bunch of physics courses in college.