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How fast was the Apollo 11 rocket going?
Short answer
The Saturn V gained speed stage by stage: after two and a half minutes, the first stage took Apollo 11 to an altitude of 61 kilometers, flying at 8,000 km/h. In Earth orbit it reached 7,791.2 meters per second, and after the burn toward the Moon, 10,841 meters per second.

On the ground, thrust counts, not speed
At liftoff, speed is almost nil, but force is enormous: the first stage's five F-1 engines delivered 7.5 million pounds of thrust combined and burned 13 tons of fuel per second. That very ratio explains why a rocket leaves the launch tower very slowly and only truly accelerates once it reaches thin air. After two and a half minutes, the first stage had taken Apollo 11 to an altitude of 61 kilometers, where the stack continued flying at 8,000 kilometers per hour.
In Earth orbit
The second and third stages then took the stack to an orbit at an altitude of 191.1 kilometers. Upon entering orbit, Apollo 11 was flying at 7,791.2 meters per second, and circled the planet for about three hours. That phase wasn't waiting time, but checking time: the crew and ground control verified systems, pressure, navigation and the orbit before reaching the point where a return would have required considerable effort.
The leap onto the lunar route
For the trip to the Moon, the third stage's engine fired again and accelerated the stack over six minutes to 10,841 meters per second; at shutdown, it was flying at an altitude of 320.9 kilometers, and the speed deviated from the plan by only 1.6 feet per second. That was the fastest stretch of the outbound trip; afterward, Earth's gravity began acting against the spacecraft and continuously slowed it down, until the Moon's pull became dominant. In rockets, mass and speed are always linked: more cargo means less speed, and vice versa.
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