Relativistic Accelerated Linear Motion Type 1 Simple Linear Acceleration From Rest to Any Given Distance in Space PHP Program by Jay Tanner
Time Units Symbol	Distance in Light Time Units	Acceleration G−Factor (1.0 = Earth)
Double-Click Within Text Area to Select ALL Text --------- SCENARIO: A body accelerating along a straight line in space. Starting from rest, the body accelerates at a given G factor until reaching distance D, at which point all acceleration stops and the body has reached the final relativistic coasting speed β. This program computes the time interval differences for both the moving body and the home clock due to the effect of relativistic time-dilation during the accelerated motion and the final speed achieved at the end. ===================================== UNITS, TARGET DISTANCE D AND G FACTOR For Time Units = Years and Dist Units = Light Years D = 1.0 Light Year and G = 1.0 (1.0 = Earth) ------------------------------------------------------------------- Home clock time taken to accelerate to distance D light years T = 1.71389327530758973524960168908668 years = 89wk 2d 23h 59m 18s ------------------------------------------------------------------- Body clock time taken to accelerate to distance D light years t = 1.29362796738030510096836619956539 years = 67wk 3d 11h 56m 33s ------------------------------------------------------------------- Time difference (Home Clock − Body Clock) T − t = 0.42026530792728463428123548952129 year = 21wk 6d 12h 02m 44s ------------------------------------------------------------------- Final relativistic speed at endpoint distance D light years β = v/c = 0.87056440674998024089642023962833 = 939559116.0487981243311718 km/h = 583814968.1166973153851331 mi/h When the acceleration stops, this is the final constant speed at which the body will then coast through space forever.
This program helps to study the effect of relativistic time- dilation at different linear acceleration rates and distances. The body can be a spacecraft or any body in accelerated linear motion through space. When the body represents a spacecraft, then times in years and distances in light years (default) may be the most practical units to apply. The home clock refers to a clock in the home (lab) rest frame. The body clock refers to a clock that travels with the body. According to relativity, the relatively moving body will measure less clock time than the relatively static home clock during the motion.
To generally apply the following relativistic equations, distances must be reckoned in related light-time units according to the simple rules given below.  This allows for simplified relativistic motion equations expressed in terms of basic space-time geometry. The distance units and the value of S used depends on the choice of time units used in the computations. IN THE COMPUTATIONS: c  =  Speed of Light  =  299792458 m/s g  =  Standard gravitational acceleration rate on Earth;  =  9.80665 m/s²  =  G factor 1.0 a  =  General acceleration rate in m/s;  =   g × (G factor) G  =  G factor  =  a/g 1 AU = 149597870700 m 1 Standard (Julian) Year  =  31557600 seconds;  =  365.25 days If time is to be reckoned in seconds, then distances must be reckoned in light seconds. S  =  1  =  Seconds per second 1 Light Second  =  299792458 m  =  0.002003988804100004 AU If time is to be reckoned in minutes, then distances must be reckoned in light minutes. S  =  60  =  Seconds per minute 1 Light Minute  =  17987547480 m  =  0.1202393282460002 AU If time is to be reckoned in hours, then distances must be reckoned in light hours. S  =  3600  =  Seconds per hour 1 Light Hour  =  1079252848800 m  =  7.2143596947600137 AU If time is to be reckoned in days, then distances must be reckoned in light days. S  =  86400  =  Seconds per day 1 Light Day  =  25902068371200 m  =  173.1446326742403293 AU If time is to be reckoned in weeks, then distances must be reckoned in light weeks. S  =  604800  =  Seconds per week 1 Light Week  =  181314478598400 m  =  1212.0124287196823049 AU If time is to be reckoned in years, then distances must be reckoned in light years (default). S  =  31577600  =  Seconds per standard (Julian) year of 365.25 days 1 Light Year  =  9466726321740800 m  =  63281.1568603482803449 AU
Program by Jay Tanner Revised: Thursday, January 01, 1970 at 12:00:00 AM UTC