Note: Any velocity mentioned in this question is the velocity relative to the earth.
Warning: Base your answers on formal proof and facts, moon images aren't really science...
According to this book page in Astronomy: A Physical Perspective By Marc Leslie Kutner:
For an object going in a circular orbit around a planet the gravitational force must provide the acceleration for circular motion. If we solve this to get the orbital velocity, we get
This means that we need that velocity to escape from the earth if we follow the earth's orbit.
Another way is to send a rocket straight up as shown in this answer:
To leave the earth, you do NOT need to go at that speed of 28,000 km/h. You simply have to be going that fast when you reach that altitude. Given enough time, something travelling at 1 m.p.h can leave the earth, or the solar system. It needs enough power so that when it stops thrusting, it's current speed exceeds it's current height's orbital velocity.
This helps explain why rockets are almost always launched east - due to the spin of the earth, they get a head start on their orbital velocity. At the Equator, you get a free boost of almost 1000 miles/hour (less so as you travel away from the equator).
So, is it possible to get to the power needed to thrust a rocket so that it can escape earth?
For the orbital way of escaping: Wikipedia mentions a velocity of
27,870 km/h which is sufficient if you count in mathematical errors, however, unless I'm incorrect I think such speeds have only been achieved by NASA. Are there any other companies that achieved this speed? And how are such speeds measured?
Someone is trying to convince me that the maximum velocity ever reached was
3,529.6 km/h by the Lockheed SR-71 Blackbird and thus space exploration does not exist because nobody has ever been there, how can I prove him wrong?
Are there any formal proofs or facts that aren't supported solely by NASA?