Time dilation effects become noticeable when an object travels at speeds approaching a significant fraction of the speed of light. According to the theory of special relativity, time dilation occurs as a result of relative motion between observers.
The equation that relates time dilation to relative velocity is given by the Lorentz factor:
γ = 1 / √(1 - v^2/c^2)
Where: γ is the Lorentz factor, v is the velocity of the moving object, and c is the speed of light in a vacuum.
To determine at what velocity time dilation effects become noticeable, we need to consider a specific level of noticeable difference. Let's say a time dilation factor of at least 10% is considered noticeable. In this case, we can set γ = 1.1 and solve for v.
1.1 = 1 / √(1 - v^2/c^2)
Squaring both sides:
1.21 = 1 / (1 - v^2/c^2)
Rearranging the equation:
1 - v^2/c^2 = 1 / 1.21
v^2/c^2 = 1 - 1/1.21
v^2/c^2 = 0.826446
v^2 = 0.826446 * c^2
v = √(0.826446 * c^2)
v ≈ 0.908c
Therefore, to observe noticeable time dilation effects, you would need to travel at approximately 0.908 times the speed of light (about 90.8% of the speed of light). At this velocity, time would appear to be dilated by a factor of 1.1 compared to a stationary observer.