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You did what now?
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I doesn't seem like catching up to me because catching up implies speed was increased to intercept, not distances were different followed by intercept.
If I fire a gun nearly perfectly straight up, run forward 10 feet and catch the bullet in my shoulder it wouldn't feel right to say that I fired, ran fast enough to catch up to my bullet and shot myself.
You fire a bullet and it accelerates downward at 9.8m/s2 until it gets to some terminal velocity. It moves forward at some velocity with a braking acceleration that's non-linear and gross. Result is a downward motion in a basically parabolic arc.
The plane, however, is accelerating downward faster than the bullet because of thrust, and also accelerating forward.
By the time drag has essentially stopped the bullet the plane is underneath it.
When phrased as "caught up" it makes it sound like the plane went as fast as the bullet, when the plane had a top speed of mach 1 and the bullet ~mach 3. They just took different paths.