Carbon-15 has a half-life of 5730 years. As observed from Earth, what would the half-life of carbon-15 be if it traveled through space at 25% of the speed of light, relative to Earth?
time for the carbon would be dilated, or slowed by the usual relativistic factor. So, multiply the half-life by that.
To determine the half-life of an isotope when it is traveling at a fraction of the speed of light, we need to account for time dilation. The time dilation formula we will use is:
t' = t * √(1 - v^2/c^2)
Where:
t' is the observed half-life from Earth
t is the half-life of carbon-15 at rest (5730 years)
v is the relative velocity of carbon-15 (25% of the speed of light)
c is the speed of light (299,792,458 meters per second)
First, we need to convert the velocity to meters per second:
v = 0.25 * c
v = 0.25 * 299,792,458 m/s
v = 74,948,114.5 m/s
Now we can calculate the observed half-life:
t' = t * √(1 - (v/c)^2)
t' = 5730 * √(1 - (74,948,114.5 / 299,792,458)^2)
t' = 5730 * √(1 - 0.062503779)
t' = 5730 * √(0.937496221)
t' = 5730 * 0.96842
t' ≈ 5,541.23 years
Therefore, the observed half-life of carbon-15, as seen from Earth when it travels at 25% of the speed of light relative to Earth, would be approximately 5,541.23 years.