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During the First World War, a French pilot flying at about two kilometres altitude noticed a small object moving beside his face and grabbed it. It turned out to be a German bullet. Near the end of its flight the bullet had slowed to roughly the plane's speed and was travelling the same way, so relative to the pilot it was almost stationary and harmless.
A moving object's capacity to hurt or be caught depends only on its velocity relative to you, not its speed over the ground. Switch the frame of reference to the observer's own motion and many paradoxes dissolve.
In a vacuum, a bullet fired straight up would return to the muzzle at its launch speed, several hundred metres per second. In air, drag grows with speed until it balances gravity. The returning bullet therefore reaches only a terminal velocity that is a small fraction of its muzzle speed, so the naive vacuum calculation badly overstates its energy.
What a scale reads is not the pull of gravity but the force the floor must exert to accelerate you along with it. In a lift accelerating upward, that force exceeds gravity and you weigh more. Accelerating downward you weigh less, and in free fall the support force and your apparent weight drop to zero.
In Jules Verne's cannon-launched trip to the Moon, the passengers feel weightless only at the point where Earth's and the Moon's gravity cancel. Perelman corrects this: once the projectile leaves the barrel, it and everything inside it fall under gravity at the same acceleration, so nothing presses on anything else. The travellers would therefore be weightless for the whole unpowered flight, not only at the neutral point.