Question

How can two observers who are moving relative to each other at different speeds come up with the same result for both measurements (e.g. length)?

Answer 1

When two observers are moving relative to each other at different speeds, they will generally perceive different measurements due to the effects of relative motion. However, under certain circumstances, they can still arrive at the same result for certain measurements, such as length. This is possible when they take into account the principles of special relativity, which describes how measurements of space and time are affected by relative motion.

In special relativity, there are two key concepts that help reconcile measurements between observers in relative motion: time dilation and length contraction.

1. Time Dilation: According to special relativity, time is not absolute but rather relative to the observer's frame of reference. When two observers are moving relative to each other at different speeds, they will perceive time differently. The observer in the faster-moving frame will measure time to be dilated or stretched out compared to the slower-moving observer. This means that processes and events appear to occur more slowly for the faster-moving observer from the perspective of the slower-moving observer.

2. Length Contraction: Similarly, special relativity predicts that the length of an object in motion appears contracted or shortened along the direction of its motion when observed by an outside observer. This effect is known as length contraction. The amount of contraction depends on the relative velocity between the two observers.

To ensure that both observers obtain the same result for a measurement, such as length, they must take into account the effects of time dilation and length contraction. By applying the appropriate equations and transformations from special relativity, they can arrive at a consistent value for the measured length.

It's important to note that the principle of relativity states that the laws of physics should be the same for all observers in inertial (non-accelerating) frames of reference. So, while measurements of length or other quantities may differ between observers in relative motion, they can still agree on certain results by considering the effects of time dilation and length contraction.