- give a direct answer to my question
- maybe point me in the right direction as to where I should be looking for
the math or answer to my question
- tell me to go away as this is a serious group

The question is:
If I have an object such as an iron ball 3 inches in diameter, wiehging 1
kilogram that is propelled upwards at the speed of light (or close), from a
stationary position with the force of gravity acting against it. How long
should it take to reach it's maximum velocity (close to light speed). I hope
this makes sense.

Thank you David, you have helped me to ask the question I was trying to form
originally.

Would you know a formula or where I should be looking to find out more
information about an accelerating objects' starting speed and its final
speed over a given distance.

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form

As long as you stay far away from c, you can always use Newton:
the relationship between velocity and acceleration
v=a * t + v0
the relationship between distance, velocity, and acceleration
s=1/2 * a * t^2 + v0 * t +s0

Since you have established that you are only interested in a differential
measurement, s0 can be zero.

If you want to work closer to c, you'll need to look at energies... a much
more complicated process. Especially when you consider that the moving
observer, and the stationary observer will not agree on the rate of speed.