Expert: Daniel Mazur - 6/1/2005

Question
My sixth grade students wondered how fast the liftoff of the
space shuttle has to be to obtain orbit?

Please respond as soon as possible, since school will be out on
June 6th.

Thank you!

Answer
For a sixth grader a satisfactory answer might be about 8 km/s. That is the same speed the rocket needs to stay at the lowest orbit. It uses some unrealistic assumptions: no air friction, the speed is given to the rocket instantaneously and the velocity vector is horizontal (not vertical). It is more like shooting a bullet from a rifle.

How to solve:
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The method to solve the problem involves the centrifugal force -m*v^2/r and Newtonean gravitational force -G*m*M/(r^2). When these two are equal then the speed

v = square_root(G*M/r)

Here G is the gravitational constant, M is the mass of Earth and R is the radius of desired orbit. I have used the equatorial Earth radius to obtain the 8 km/s above. I met these laws and this calculation in my ninth grade, so I should not think giving the derivation to your sixth-graders would be apropriate.


The reality of this problem is very complex. The rocket starts from zero speed, then it accelerares. Even the acceleration changes with time: the air friction to overcome changes with altitude (boosts acceleration) and a the rocket becomes lighter and lighter as it burns its fuel (helps the acceleration even more). Giving the liftoff speed of a real rocket is therefore not sensible, I am afraid.

I hope this helps you in your work. Best luck to you, Mrs. Andreas, and do not hesitate to rate my answer here at allexperts.com.

Thank you.

Daniel Mazur