(I) Repeat Problem 19 using the Lorentz transformation and a relative speed v = 1.60 x 10⁸ m/s, but choose the time t to be (a) 3.5μs and (b) 10.0 μs .

Verified step by step guidance

1. Identify the given values: relative speed v = 1.60 x 10^8 m/s, and times t = 3.5 \(\text{μs}\) and t = 10.0 \(\text{μs}\). Convert these times into seconds for consistency in units.
2. Recall the Lorentz transformation equations for time: t' = \(\gamma\) (t - \(\frac{vx}{c^2}\)), where \(\gamma\) is the Lorentz factor given by \(\gamma\) = \(\frac{1}{\sqrt{1 - \frac{v^2}{c^2}\)}}, c is the speed of light (approximately 3.00 x 10^8 m/s), and x is the position in the original frame.
3. Calculate the Lorentz factor (\(\gamma\)) using the given relative speed v.
4. Substitute the values of t, v, and \(\gamma\) into the Lorentz transformation equation for time to find t' for each case: (a) t = 3.5 \(\text{μs}\) and (b) t = 10.0 \(\text{μs}\). Assume x = 0 if the position is not specified.
5. Interpret the results to understand how time dilation occurs due to high relative speeds close to the speed of light.

Key Concepts

Lorentz Transformation

The Lorentz transformation is a set of equations in special relativity that relate the space and time coordinates of two observers moving at a constant velocity relative to each other. It accounts for the effects of time dilation and length contraction, ensuring that the speed of light remains constant in all inertial frames. This transformation is essential for analyzing scenarios involving high speeds, such as the one presented in the problem.

Time Dilation

Time dilation is a phenomenon predicted by Einstein's theory of relativity, where time is observed to pass at different rates for observers in different inertial frames. Specifically, a moving clock ticks slower compared to a stationary clock as perceived by an observer. This concept is crucial for understanding how time intervals, such as the given 3.5 μs and 10.0 μs, will be perceived differently depending on the relative speed of the observers.

Relative Velocity

Relative velocity is the velocity of one object as observed from another object. In the context of special relativity, it is important to consider how the speed of light remains constant regardless of the relative motion of observers. The problem specifies a relative speed of 1.60 x 10⁸ m/s, which is significant enough to invoke relativistic effects, making it necessary to apply the Lorentz transformation to accurately analyze the situation.