Is a plane fast enough to get to space? Well, unfortunately for space exploration, no, it isn’t. Let’s check with a few calculation.
Assumptions and definitions
First, let’s make an assumption: our plane P is flying in the void: there is no air slowing it down. Of course, this assumption is very wrong, but we only care about speed here. Air friction only slows P down. So, if P doesn’t go into space without an atmosphere, it won’t either if we add an atmosphere.
Let’s consider a reference object O orbiting the Earth at the same altitude r as our plane. Of course, we can neglect the mass of P and O, which is several orders of magnitude lesser than the mass of the Earth.
Let vO be the orbital speed of O and vP be the speed of P. If vP > vO, then P will spiral away from Earth and end up in space. If, on the other hand, vP < vO, then P will spiral down until it crashes.
Because O is orbiting on a circular trajectory, vO and r are correlated by the following formula:
vO = √(GM/r)
Where G is the gravitational constant and M the mass of the object we’re orbiting around.
In our case, we are orbiting around the Earth and
GM = 3.99×1014 m3s-2
By using the SI units and approximating 3.99 with 4, we end up with the following relation:
vO = 2×107 r-1/2
The radius of the Earth is included in r, so we must add 6 371 km to go from altitude to r. 6 371 km is actually two order of magnitude higher than any altitude you could fly at, so let’s round it to 6,400 km and be happy with it.
So, in order to orbit at 6 400 km = 6 400 000 m from the center of Earth, we would need to go at
vO = 2×107 r-1/2 = 2×107 (6 400 000)-1/2 m/s = 2×107/2530 m/s = 7.9×103 m/s = 1.8×104 mph
That’s the speed relative to the center of the Earth, not to its surface.
So, does our poorly modeled plane fly?
Our plane P, is flying at 600 mph relative to the surface. If we suppose we’re flying above the equator, we can reach a speed of 1600 mph relative to the center; that still is an order of magnitude below the speed we need to reach to orbit at such a low altitude.
vP << vO
Our plane would crash. Fast.
Thank goodness, planes are safer than that. But how come?
Well, as we’ve just seen, the speed of the plane is not what keeps it flying. But we made a very strong assumption in our opening section: we neglected the atmosphere.
Air certainly slow our plane down. But we’ve also built it so that, when moving through air, our plane tends to go up thanks to lift.
So we have two opposing effects, here: the plane isn’t going fast enough to escape the Earth attraction; but the lift pulls up the plane hard enough that it keeps flying. But lift depends, among other things, on the density of air around the plane: the higher the planes fly, the weaker lift is. So, all in all, the plane flies in a layer of air at the same pressure, which in turn follows approximately the surface of the Earth.