neufer wrote: ↑
Thu May 10, 2018 2:36 pm
I've always liked that diagram. As a person who has learned just enough physics to be a danger to myself and others ... it occurred to me it would be fun to calculate just what speed it would take to get exactly once around the Earth and to land at your feet, say, just 1 foot behind you. (We must be precise ... we don't want the cannonball to hit us in the back of the head, or even in the derriere.)
So, if you fired an 8 kg cannonball at 30 degrees up from horizontal (let's just fire it from the ground, not a tower) and you imparted to it a velocity of 500 m/s, it would arc up and come back down, landing about 10,000 m away. (That's from real experimental data http://www.desertrats.org.uk/equipartillery.htm
Starting with a "no-air-resistance" approach, there is a nice calculator for this at: http://www.convertalot.com/ballistic_tr ... lator.html
. Playing around with this, I get to 21.306 km/s muzzle velocity, at 30 deg elevation, would land a distance of 40,075 km away, which is the circumference of the Earth.
Fun enough to that point. What will be the effect if I could make a calculator that included air resistance? Obviously, a cannonball shot at 21 km/s would not make it around the Earth once. Air resistance would slow it down. But by how much? And if we were to try to overcome that, by putting larger initial velocities into this hypothetical calculator, how high would we have to go to overcome air resistance and then get a range of 40,075 km? It is possible that no velocity will work no matter how high, because air resistance apparently grows as the square of a projectile's velocity. And actually, there appears to only be a certain range of velocities within which this quadratic relation holds, so I don't know what goes on at super-high velocities (other than immense heating).
A nice calculator that includes air resistance and gives its result with graphical output is at: http://dynref.engr.illinois.edu/afp.html
None of the above discussion actually takes into account Newton's point in his cannonball thought experiment, that the Earth is curved, not flat. This of course helps, and as Newton pointed out from the diagram, it can produce infinite-length flight paths. However, if a projectile does not get out of the atmosphere, the drag from the air will defeat this. In his thought experiment, the projectile keeps its speed, whereas within the atmosphere, this would not occur.