- #1
spica
- 5
- 0
A simple problem with a constant acceleration, ignoring mass of the fuel.
The velocity of a rocket, which moves withe a constant acceleration g, is equal:
v(t) = gt
but I want to keep the const acceleration inside the rocket, not in the absolute sense.
An acceleration has a dimension: L/T^2 [m/s^2];
and in a moving rocket the meter is contracted gamma times,
and a time is dilated gamma times too,
so finally the absolute acceleration must be: ##g / \gamma^3##
therefore:
##dv=\frac{g}{\gamma^3}dt##
which can be easili integrated:
##\gamma^3dv = gdt\to\, v\gamma = gt\to\, \frac{v}{\sqrt{1-v^2}}=gt##
finally:
##v(t) = \frac{gt}{\sqrt{1+(gt)^2}}##
Is this a correct speed of the relativistic rocket?
The velocity of a rocket, which moves withe a constant acceleration g, is equal:
v(t) = gt
but I want to keep the const acceleration inside the rocket, not in the absolute sense.
An acceleration has a dimension: L/T^2 [m/s^2];
and in a moving rocket the meter is contracted gamma times,
and a time is dilated gamma times too,
so finally the absolute acceleration must be: ##g / \gamma^3##
therefore:
##dv=\frac{g}{\gamma^3}dt##
which can be easili integrated:
##\gamma^3dv = gdt\to\, v\gamma = gt\to\, \frac{v}{\sqrt{1-v^2}}=gt##
finally:
##v(t) = \frac{gt}{\sqrt{1+(gt)^2}}##
Is this a correct speed of the relativistic rocket?