How to Calculate Time Dilation in Special Relativity
What is the time taken for the ball bearing to fall according to the passenger?
To determine the time measured by the passenger, we can use the concept of time dilation in special relativity. Given that the Gautrain is traveling at 0.5 c (half the speed of light), we can calculate the time dilation factor using the Lorentz factor: γ = 1 / √(1 - v^2/c^2), where v is the speed of the Gautrain and c is the speed of light. Once we have the time dilation factor, we can multiply it by the time it would take for the ball bearing to fall in the stationary frame (which is approximately 1.46 seconds) to obtain the time measured by the passenger, which is approximately 3.17 seconds.
Answer:
The time taken for the ball bearing to fall according to the passenger is approximately 3.17 seconds.
Time dilation is a fascinating concept in special relativity where time intervals between events can differ for observers in relative motion. In the scenario given, the passenger on the Gautrain experiences time dilation due to the high speed of the train. This means that the passenger perceives time differently compared to a stationary observer.
By applying the Lorentz factor and understanding the effects of relativistic speeds, we can calculate the time measured by the passenger accurately. It is essential to consider the differences in perception of time between observers in motion and those at rest to grasp the concepts of time dilation fully.
In conclusion, mastering the calculations of time dilation in special relativity can provide valuable insights into the nature of time and space in our universe.