The effects of time dilation become apparent at speeds that are a significant fraction of the speed of light. According to special relativity, time dilation occurs as a result of relative motion between observers. As an object's velocity approaches the speed of light (denoted as "c"), the effects of time dilation become increasingly pronounced.
The formula for time dilation in special relativity is given by the equation:
Δt' = Δt / √(1 - v^2/c^2)
where Δt' is the time interval experienced by the moving object, Δt is the time interval measured by a stationary observer, v is the velocity of the moving object, and c is the speed of light.
To illustrate the point at which time dilation becomes noticeable, let's consider an example. Suppose you have two observers: one is stationary (v = 0), and the other is moving at a significant fraction of the speed of light (v ≈ c). If we use the equation above, the denominator approaches zero as v approaches c, resulting in a time dilation factor approaching infinity. This means that as an object's velocity approaches the speed of light, time dilation becomes more and more significant, and the moving object experiences time passing more slowly compared to the stationary observer.
In practical terms, noticeable time dilation effects become evident at speeds comparable to a significant fraction of the speed of light, such as 0.9c or 0.99c. At such velocities, the time dilation factor can be substantial. For example, at 0.9c, the time dilation factor is about 2.29, meaning that for every second experienced by the stationary observer, only about 0.44 seconds would pass for the moving object.
However, it's important to note that significant relativistic effects are typically observed at speeds much closer to the speed of light, as the time dilation factor increases rapidly as the velocity approaches c. For everyday velocities (compared to the speed of light), the effects of time dilation are negligible and not noticeable.
Overall, the effects of time dilation become apparent and significant when an object's velocity approaches a substantial fraction of the speed of light, leading to the fascinating phenomena predicted by special relativity.