According to the theory of relativity, time dilation occurs when an object is moving relative to an observer at a significant fraction of the speed of light. The faster the object moves, the slower time appears to pass for the moving object relative to a stationary observer.
To calculate the relativistic speed of the spaceship as measured by the onboard clock, we can use the time dilation formula. The formula for time dilation in special relativity is:
t' = t / √(1 - v^2/c^2)
Where: t' is the time measured by the moving object (onboard clock) t is the time measured by a stationary observer (in this case, let's assume it's the time on Earth) v is the velocity of the spaceship relative to the stationary observer c is the speed of light in a vacuum
In this scenario, the spaceship is traveling at 90% the speed of light, which can be written as v = 0.9c. We can now plug in the values and calculate the relativistic speed:
t' = t / √(1 - (0.9c)^2/c^2) = t / √(1 - 0.81) = t / √(0.19) = t / 0.4359
Therefore, the relativistic speed of the spaceship as measured by the onboard clock is approximately 0.4359 times the speed of light.