How fast would a bullet have to travel to effectively beat gravity?

I was reading and saw this post

How does gravity affect bullets?

So, my question is: Assuming the hypothetical planet was identical to earth is size and gravity but a total vacuum and completely "flat" in such there are no hills, mountains, etc... obstacles in the way, how fast would a bullet have to travel to essentially beat gravity and continue to stay at the same height and travel around the world to essentially hit the firearm in which it was shot out of? And roughly, how long would it take for said projectile to go around the "earth" and hit the firearm?

If it's not clear by what I'm saying, I don't mean beat gravity as in fly out to outer space like a rocket, or if there was zero gravity it would just fly off because nothing is keeping it here. I assume there has to be a certain speed at which you can fire a firearm from about 5 feet in the air to which gravity affects the bullet at the same rate that the curvature of the "earth" happens. Gravity would pull it down at the same rate as the bullet would "rise" above the ground making it stay at 5 feet constantly around the planet.

You're looking for the orbital velocity. This is given by equating the centrifugal force with the gravitational force:

$${mv^2 \over {r}} = {Gmm_{e} \over {r^2}}$$

• $m$ is the mass of the bullet, or projectile, which cancels out (its mass doesn't matter - all projectiles at a given speed and height follow the same orbit).
• $r$ is the radius of the orbit - basically, the radius of the Earth; $6.371 \times 10^6$m.
• $G$ is the gravitational constant that tells us how strong gravity is; $6.67 \times 10^-{11} m^3 kg^{-1} s^{-2}$
• $m_e$ is the mass of the Earth; $5.972 \times 10^{24}kg$

Transposing the equation: $$v = \sqrt {Gm_e\over{r}}$$ Sticking in the values (and approximating a bit), gives $$v \approx 8 kms^{-1}$$ Which is about 18,000 mph, or Mach 23.

I think bullets are usually a bit slower than this...

• (Just for comparison purposes, typical values for initial bullet speed is on the order of just 1000-2000 mph) – JeopardyTempest Jan 31 '18 at 10:54
• The fastest bullet known is roughly 4100ish FPS. So, only need to increase that by about 6 1/2 times to get the required 2600o FPS. I believe in science to get this done. I"m assuming on a planet like Pluto it would almost be laughably easy just because of how small the planet is itself. It would almost be hard to find a projectile slow enough to fire out of a firearm to travel slow enough? – Tauqmuk Jan 31 '18 at 13:45
• For Pluto, the orbital velocity near the surface is about 2700 fps. I guess there's a few rifles could hit that... – Oscar Bravo Jan 31 '18 at 14:26