In classical mechanics, the principle of relativity states that the laws of physics are the same for all observers in inertial reference frames. This means that the relative motion of two observers moving with constant velocity relative to each other should not affect the fundamental physical laws and measurements.
To derive the clock time difference between two observers moving with constant velocity relative to each other using classical mechanics, we can consider the concept of time dilation.
Time dilation is the phenomenon where the elapsed time measured by a clock in motion relative to an observer differs from the elapsed time measured by a clock at rest relative to the observer. In classical mechanics, time dilation occurs due to the relative motion of the observers.
Let's consider two observers, A and B, where A is at rest and B is moving with a constant velocity relative to A. Observer A has a clock at rest, and observer B has a clock in motion.
According to the principle of relativity, observer B should observe the same physical laws as observer A. However, due to their relative motion, observer B will perceive observer A's clock to be running slower compared to their own clock.
The time dilation factor, γ, can be expressed as:
γ = 1 / sqrt(1 - (v^2 / c^2))
where:
- v is the relative velocity between the observers.
- c is the speed of light.
By multiplying the elapsed time measured by observer A (Δt_A) by the time dilation factor γ, we can calculate the elapsed time measured by observer B (Δt_B):
Δt_B = γ * Δt_A
Since observer B's clock is considered to be moving relative to observer A, Δt_B will be greater than Δt_A. This means that observer B's clock will show a larger elapsed time compared to observer A's clock.
This derivation of the clock time difference between two observers moving with constant velocity relative to each other is based on classical mechanics. However, to accurately describe time dilation and other relativistic effects, we need to employ the framework of special relativity, which incorporates the constancy of the speed of light and the relativistic effects of time and space.