What is the minimum speed with which you can launch the projectile and have it complete an orbit of the earth?

Question

and What happens if you launch the projectile with less than the minimum speed

Answer 1

The lowest possible orbit, just barely outside the atmosphere, would put the projectile 6.4x10^6 m from the center of the Earth, and acceleration due to gravity there is 9.7 m/s². Putting those numbers into the formula a=v²/r and solving for v gives us 7.88 km/s (17,600 mph).

If you launch the projectile from the equator, it already has an eastward velocity of 460 m/s, so you'd only have to launch it eastward at a 7.42 km/s relative to the ground.

These calculations ignore the air resistance the projectile would encounter on its way up.

If you launch the projectile at less than 7.42 km/s it will travel a distance shorter than the circumference of the Earth.

Escape velocity is not the answer. That would tell you the maximum speed a projectile could be launched and have it eventually return to Earth, not the minimum.

EDIT: I wrote a computer simulation this morning to calculate the maximum distance (in degrees longitude) the projectile would travel before returning to Earth, given various inputs of launch velocities. Below is a summary of the results.
5.0 -> 29°
6.0 -> 48°
7.0 -> 80°
7.5 -> 110°
7.7 -> 129°
7.8 -> 143°
7.9 -> 173°
8.0 -> 360°
Again, if you are launching eastward from the equator, you can subtract .46 km/s because the launch pad is already moving 460 m/s east.

As you can see, decreasing the launch speed just a little bit below the critical number has a dramatic effect on distance traveled. 8 km/s will get you all the way around the world but 7 km/s won't even get you 1/4 of the way around.

By the way, in addition to ignoring wind resistance, I also ignored the fact that, during the time the projectile is in the air, the Earth is rotating beneath it. If you measure the distance traveled along the Earth you get a slightly shorter answer than the actual distance traveled relative to your original launch point.

Answer 2

the speed of the lowest orbit is
escape speed / SQRT(2)

For Earth:
11.2 km/s divided by 1.4142 = 7.92 km/s
(17,820 mph)

If you don't mind your projectile crashing on its launch site after one orbit (i.e., NOT a stable orbit that decays before the 2nd orbit begins), you could get by with anything above 7.1 or 7.2 km/s (16,250 mph)

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If you launch the projectile at (or above) escape speed, it escapes Earth altogether (it does not stay in an orbit around Earth).

Answer 3

It depends on where you launch it (latitude and elevation) and what direction you want it to orbit but escape velocity as roughly 25000 miles per hour.

If it does not have sufficient speed to escape earth's gravitation, it will fall back onto the earth.