The Apparent Velocity and Acceleration of Relativistically Moving Objects

TL;DR

This paper explores how the apparent velocity and acceleration of relativistically moving objects differ from their actual values, especially as their actual speed approaches the speed of light, revealing surprising observational effects.

Contribution

It analyzes the apparent velocity and acceleration of particles moving with constant speed or acceleration, highlighting how relativistic effects influence observed motion.

Findings

Apparent velocity approaches infinity when approaching the observer at near-light speeds.

Apparent velocity approaches c/2 when receding at near-light speeds.

Results reveal counterintuitive observational phenomena in relativistic motion.

Abstract

Although special relativity limits the actual velocity of a particle to $c$, the velocity of light, the observed velocity need not be the same as the actual velocity as the observer is only aware of the position of a particle at the time in the past when it emits the detected signal. We consider the apparent speed and acceleration of a particle in two cases, one when the particle is moving with a constant speed and the other when it is moving with a constant acceleration. One curious feature of our results is that in both cases, if the actual velocity of the particle approaches $c$, then the apparent velocity approaches infinity when it is moving toward the observer and $c/2$ when it is moving away from the observer.