To solve for velocity (v) when you have the initial time (t₀) and the time measured in a different reference frame (t) due to time dilation, you can use the time dilation equation from special relativity:
t = t₀ / √(1 - (v²/c²))
In this equation, c represents the speed of light in a vacuum, and v is the velocity of the moving object relative to an observer.
To isolate the velocity (v) in the equation, you can follow these steps:
Square both sides of the equation: t² = (t₀ / √(1 - (v²/c²)))²
Multiply both sides by the denominator squared: t²(1 - (v²/c²)) = t₀²
Expand the equation: t² - (v²/c²)t² = t₀²
Factor out t² from the left side: t²(1 - (v²/c²)) = t₀²
Divide both sides by t₀²: (t²(1 - (v²/c²))) / t₀² = 1
Rearrange the equation: (1 - (v²/c²)) = t₀² / t²
Subtract 1 from both sides:
Multiply both sides by -c²: v² = -c²((t₀² / t²) - 1)
Take the square root of both sides: v = ±√[-c²((t₀² / t²) - 1)]
Note: The negative sign indicates that the velocity can be in the opposite direction of the observer's motion.
By plugging in the known values for t₀ and t, you can calculate the velocity (v) using this equation.