In classical mechanics, the principle of relativity states that the laws of physics should be the same for all observers in inertial reference frames. This principle allows us to derive the clock time difference between two observers moving with constant velocity relative to each other.
Let's consider two observers, Observer A and Observer B, who are moving with constant velocity relative to each other along a straight line. Without loss of generality, let's assume that Observer A is at rest, and Observer B is moving with velocity v relative to A.
Now, let's suppose there is a clock located at the position of each observer. According to classical mechanics, the time measured by each clock will depend on their relative motion. We want to determine the time difference between the two clocks as observed by each observer.
From Observer A's perspective, they are at rest, so the time measured by their own clock is the proper time, denoted as t_A. In this case, t_A represents the time experienced by Observer A.
From Observer B's perspective, they are moving with velocity v. According to classical mechanics, time dilation occurs due to relative motion. The time measured by Observer B's clock, denoted as t_B, will be dilated or stretched compared to t_A.
The time dilation factor, according to classical mechanics, is given by the equation:
t_B = t_A / sqrt(1 - (v^2 / c^2))
Where c is the speed of light in a vacuum. This equation is derived from the Lorentz transformation, which relates the coordinates and time measurements between observers in relative motion.
Therefore, using classical mechanics and the principle of relativity, we can determine the time difference between two observers moving with constant velocity relative to each other by considering the time dilation factor and the proper time measured by the stationary observer.