Relativistic Spacecraft Motion Type-1 Worksheet PHP Program by Jay Tanner of Waterloo, NY, USA

Acceleration Rate (m/s²)

Distance Value in Light Time Units

Time Units Symbol

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Double-Click Within Text Area to Select ALL Text **************************************************************** NUMERICAL RELATIVISTIC SPACECRAFT MOTION TYPE 1 WORKSHEET ############################################################################# APPLIED DATA AND DEFINITIONS: Time Units = Years Distance Units = Light Years c = Speed of Light. = 299,792,458 m/s a = Acceleration experienced on spacecraft (m/s²). g = Standard gravitational acceleration on Earth. = 9.80665 m/s² G = Accelerative G-Factor corresponding to (a). Earth G = 1.0 = a / 9.80665 The value of (K) used below depends on the time units we choose to apply to the computations. Its value equates to the number of seconds in the chosen time unit. Distances are reckoned in the chosen light time units. In other words, if the chosen time unit is years, then distances are reckoned in Light Years. If the chosen time units are days, then distances are reckoned in Light Days, etc.. ############################################################################# ========================================================== Time Units K = Seconds in Time Unit Distance Units ============== ======================== ============== Seconds 1 Light Seconds Minutes 60 Light Minutes Hours 3600 Light Hours Days 86400 Light Days Weeks 604800 Light Weeks Standard Years 31557600 (365.25 days) Light Years ============== ======================== ============== In this case, we are using Years as the time units, so the value of K = 31557600 = Seconds per Year. ***************************************************************************** THE BASIC SCENARIO and RELATED NUMERICAL COMPUTATIONS Acceleration rate experienced on the spacecraft. a = 9.80665 m/s² = 1 G = 32.17404855643044619422572178477690288713910761154855 ft/s² --------------------------------------------------- Distance to the target destination in light units. D = 1 Light Year ----------------------------------------------------------- Relativistic Acceleration Potential. This is the relativ- istic speed we would reach by accelerating at rate (a) for (K) seconds - IF - there was no speed-of-light limit (c). A = a * K / c = 1.03229527555359648173670866663363492619951099637069 --------------------------------------------------- Earth time it takes to accelerate to distance (D). T = SqRt(D² + 2*D/A) = 1.71389327530758973524960168908668464325791750867897 Years 2T = 3.42778655061517947049920337817336928651583501735795 ----------------------------------------------------------------- Proper (spacecraft) time it takes to accelerate to distance (D). t = Ln(A*((D + 1/A) + SqRt(D² + 2*D/A))) / A = 1.29362796738030512498669855786405158312492435697528 Years 2t = 2.58725593476061024997339711572810316624984871395057 --------------------------------------------- Relativistic speed achieved at distance (D). This value will always be less than 1. β = SqRt(D² + 2*D/A) / (D + 1/A) = v/c = 0.87056440674998024089642023962833978657412996214615 ----------------------------------------------------------------- Speed achieved at distance (D). This value will always be less than the speed of light (299,792,458 m/s). v = β * c = 260,988,643.34688836786976994703912899107625382056324196307605 m/s = 939,559,116.04879812433117180934086436787451375402767106707380 km/h = 583,814,968.11669731538513320293291202370314473103803230824099 mi/h *****************************************************************************

Double-Click Within Text Area to Select ALL Text ****************************************************************************** SOME GENERAL INFORMATION This program computes the basic relativistic linear motion statistics for an interstellar or intergalactic space voyage, taking into account the effects of relativistic time dilation out to millions of light years. Computations are carried out to 50 decimals. However, it can also be used for space voy- age computations within the solar system as well as for target distances of hundreds or millions of light years. The initial default values are set to compute the statistics for a voyage at 1G acceleration to a target 1 light year away. This is a common problem often encountered in studying the basics of relativistic space voyages. The program allows the user to experiment with various acceleration, distance and time units and compute the effects of relativistic time dilation on both parties during and at each end of the voyage. The ultra-high precision is to make it easier to numerically compare Newton- ian physics to relativistic physics when working with ordinary or smaller distances. When working with only double precision, such numerical com- parisons are hard to make very accurately. In reality, at this point in time, we cannot yet make the kind of engines we require to make the kind of voyages so easily simulated on paper. But we can at the least compute the fantasy while we await the required technology to arise, if ever. The time dilation problem makes manned voyages to any extremely remote places very impractical. It could take hundreds or even thousands of Earth years to make a round trip voyage even to nearby stars, whether manned or robotic. Also see this ref: https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html CAVEAT: In considering the grievous logistics involved with such voyages, our civili- zation will probably be annihilated by a cosmic or anthropological calamity before we develop such space travel technology, but one can dream - ad astra. ******************************************************************************

PHP Program by Jay Tanner - 2025 v1.00 - Revised: 1970-January-01-Thursday at Local Time 12:00:00 AM (UTC−05:00)