<?php /* ########################################################################### RELATIVISTIC SPACECRAFT MOTION TYPE 1 NUMERICAL WORKSHEET AUTHOR : Jay Tanner - 2025 LANGUAGE : PHP v8.2.12 LICENSE : Public Domain This program computes the basic relativistic linear motion statistics for an intestellar or intergalactic space voyage, taking into account the effects of relativistic time dilation. Computations are carried out to 50 decimals. However, it can also be used for travel computations within the solar system as well. 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. ########################################################################### */ ob_start(); // Initialize output buffer. $cYear = date('Y'); // --------------------------------------------------------------- // Define the program cookie name and set it to expire in 30 days. $CookieName = 'Relativistic-Spacecraft-Motion-Type-1-Worksheet'; $ExpiresIn30Days = time() + 30*86400; // --------------------------------- // Define PHP program and HTML info. $_AUTHOR_ = "Jay Tanner - $cYear"; $_PROGRAM_VERSION_ = 'v1.00 - '; $at = "at Local Time "; $LTC = "UTC"; $_SCRIPT_FILE_PATH_ = Filter_Input(INPUT_SERVER, 'SCRIPT_FILENAME'); $_REVISION_DATE_ = $_PROGRAM_VERSION_ .'Revised: '. date("Y-F-d-l $at h:i:s A ($LTC", FileMTime($_SCRIPT_FILE_PATH_))."−05:00)"; $_BROWSER_TAB_TEXT_ = "Relativistic Spacecraft Motion Type 1"; $_INTERFACE_TITLE_ = "<span style='font-size:14pt;'>Relativistic Spacecraft Motion Type-1 Worksheet</span><br><br><span style='font-size:11pt;'>PHP Program by Jay Tanner of Waterloo, NY, USA</span>"; /* ------------------------------------- Define main TextArea text and background colors and HTML table row span. If an error is reported, then these colors will change internally to red/white. */ $TxColor = 'black'; $BgColor = 'white'; // --------------------------------------------- // Do this only if [SUBMIT] button was clicked. $w = Filter_Input(INPUT_POST, 'SubmitButton'); if (!IsSet($w)) { /* ---------------------------------------------------------------------- If this program is being called externally, rather than being executed by clicking the [SUBMIT] button, and an active cookie also exists, then restore the previously saved interface settings from it. If the user leaves and comes back later, all the interface settings will be remembered and restored if the cookie was not deleted. */ $w = Filter_Input(INPUT_COOKIE, $CookieName); if (IsSet($w)) { $CookieDataString = Filter_Input(INPUT_COOKIE, $CookieName); list($a,$D,$TimeUnitsSymbol) = Preg_Split("[\|]", $CookieDataString); } else /* ----------------------------------------------------------- If there is no previous cookie with the interface settings, then set the initial default interface startup values and store them in a new cookie. */ { $a = '9.80665'; // Acceleration Rate (m/s²) $D = '1'; // Distance in Light Years $TimeUnitsSymbol = 'Y'; // Time Units Symbol // ------------------------------------------- // Store current interface settings in cookie. $CookieDataString = "$a|$D|$TimeUnitsSymbol"; SetCookie ($CookieName, $CookieDataString, $ExpiresIn30Days); } // End of else {...} } // End of if (!isset(_POST['SubmitButton'])) /* ------------------------------------------ Read values of all interface arguments and set any empty arguments to default values. */ $w = Filter_Input(INPUT_POST, 'SubmitButton'); if (isset($w)) { $a = trim(Filter_Input(INPUT_POST, 'a')); $a = Str_Replace('-', '', $a); $D = trim(Filter_Input(INPUT_POST, 'D')); $D = Str_Replace(' ', '', $D); $D = Str_Replace('-', '', $D); $D = (StrPos($D, '.') === FALSE)? "$D.0" : $D; $D = RTrim(RTrim(Str_Replace(',', '', $D), '0'), '.'); $TimeUnitsSymbol = trim(Filter_Input(INPUT_POST, 'TimeUnitsSymbol')); $TimeUnitsSymbol = StrToUpper(substr($TimeUnitsSymbol,0,1)); // ------------------- // Set default values. if ($a == '') {$a = '9.80665';} if ($D == '') {$D = '1.0';} if ($TimeUnitsSymbol == '') {$TimeUnitsSymbol = 'Y';} // ------------------------------------- // Store interface argument in a cookie. $CookieDataString = "$a|$D|$TimeUnitsSymbol"; SetCookie ($CookieName, $CookieDataString, $ExpiresIn30Days); } /* --------------------------------------------- Optionally check ($a) value for validity. If error, set error flag and message values. */ $ErrFlag = FALSE; $ErrMssg = ''; // ------------------------------------------------- if (StrPos('SMHDWY', $TimeUnitsSymbol) === FALSE) { $ErrFlag = TRUE; $ErrMssg = "Bad time units symbol: $TimeUnitsSymbol The time units symbols are: S Seconds M Minutes H Hours D Days W Weeks Y Years The symbols are NOT case-sensitive."; } // -------------------- if (!Is_Numeric($D)) { $ErrFlag = TRUE; $ErrMssg = "Bad distance value:\n$D\n\nThe distance value must be a plain positive number."; } // -------------------- if (!Is_Numeric($a)) { $ErrFlag = TRUE; $ErrMssg = "Bad acceleration value:\n$a\n\nThe acceleration value must be a plain number greater than zero."; } /* ----------------------------- Set initial uniform width for tables alignment in pixels. */ $TableWidth = '780'; // ****************************************** // ****************************************** // If error was reported (TRUE), then display // the error message on a red background. if ($ErrFlag === TRUE) { $TxColor = 'white'; $BgColor = '#CC0000'; $TextArea2Text = ''; $TextArea1Text = "=== ERROR === $ErrMssg"; } else { // ********************************************************* // BEGIN MAIN COMPUTATIONS HERE IF NO ERRORS DETECTED ABOVE. // ********************************************************* // ---------------------------------- // Computation or function call code. $statistics = Relativity_101 ($a, $D, $TimeUnitsSymbol); // ******************************************* // DROP THROUGH HERE AFTER COMPUTATIONS ABOVE // TO PRINT OUT THE RESULTS OF THE OPERATIONS. // ******************************************* $TextArea1Text = " $statistics "; } // **************************** // Define TextArea2 text block. $TextArea2Text = "****************************************************************************** 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. ****************************************************************************** "; // ************************************************************************** // Determine number of text columns and rows to use in the output text areas. // These values vary randomly according to the text block width and length. // The idea is to eliminate the need for scroll-bars within the text areas // or worry as much about the variable dimensions of a text display area. // ------------ // Text Area 1. $Text1Cols = 1 + Max(Array_Map('StrLen', PReg_Split("[\n]", trim($TextArea1Text)))); if ($Text1Cols <= 78) {$Text1Cols = 78;} // Default $Text1Rows = 2 + Substr_Count($TextArea1Text, "\n"); // ------------ // Text Area 2. $Text2Cols = 1 + Max(Array_Map('StrLen', PReg_Split("[\n]", trim($TextArea2Text)))); if ($Text2Cols < 80) {$Text2Cols = 80;} // Default $Text2Rows = 2 + Substr_Count($TextArea2Text, "\n"); // ****************************************** // GENERATE CLIENT WEB PAGE TO DISPLAY OUTPUT print <<< HTML_WEB_PAGE <!DOCTYPE HTML> <HTML> <head> <title>$_BROWSER_TAB_TEXT_</title> <meta name='viewport' content='width=device-width, initial-scale=0.8'> <meta http-equiv='content-type' content='text/html; charset=UTF-8'> <meta http-equiv='pragma' content='no-cache'> <meta http-equiv='expires' content='-1'> <meta name='description' content='Numerical relativistic spacecraft motion type 1 statistics worksheet'> <meta name='keywords' content='NeoProgrammics/PHPScienceLabs'> <meta name='author' content='Jay Tanner - https://www.NeoProgrammics.com'> <meta name='robots' content='index,follow'> <meta name='googlebot' content='index,follow'> <style> BODY {color:white; background:black; font-family:Verdana; font-size:12pt; line-height:125%;} TABLE {font-size:13pt; border: 1px solid black;} TD { color:black; background:white; line-height:150%; font-size:10pt; padding:6px; text-align:center; } UL {font-family:Verdana; font-size:12pt; line-height:150%; text-align:justify;} PRE { background:white; color:black; font-family:monospace; font-size:12.5pt; font-weight:bold; text-align:left; line-height:125%; padding:6px; border:2px solid black; border-radius:8px; page-break-before:page; } DIV { background:white; color:black; font-family:Verdana; font-size:11pt; font-weight:normal; line-height:125%; padding:6px; } TEXTAREA { background:white; color:black; font-family:monospace; font-size:11.5pt; font-weight:bold; padding:4pt; white-space:pre; border-radius:8px; line-height:125%; } INPUT[type='text']::-ms-clear {width:0; height:0;} INPUT[type='text'] { font-family:monospace; color:black; background:white; font-size:13pt; font-weight:bold; text-align:center; box-shadow:2px 2px 3px #666666; border:1px solid black; border-radius:4px; } INPUT[type='text']:focus { font-family:monospace; background:white; box-shadow:2px 2px 3px #666666; font-size:13pt; border:2px solid blue; text-align:center; font-weight:bold; border-radius:2px; } INPUT[type='submit'] { background:black; color:cyan; font-family:Verdana; font-size:10pt; font-weight:bold; border-radius:4px; border:4px solid #777777; padding:3pt; } INPUT[type='submit']:hover { background:black; color:white; font-family:Verdana; font-size:10pt; font-weight:bold; border-radius:4px; border:4px solid red; padding:3pt; } // Link states MUST be set in the following order: // :link, :visited, :hover, :active A:link { font-size:10pt; background:transparent; color:#8080FF; border-radius:4px; font-family:Verdana; font-weight:bold; text-decoration:none; line-height:175%; padding:3px; border:1px solid transparent; } A:visited { font-size:10pt; background:transparent; color:DarkCyan; border-radius:4px; } A:hover { font-size:10pt; background:yellow; color:black; border:1px solid black; box-shadow:1px 1px 3px #222222; border-radius:4px; } A:active { font-size:10pt; background:yellow; color:black; border-radius:4px; } HR {background:red; height:4px; border:0px;} [title-text]:hover:after { opacity:1.0; transition:all 1.0s ease 1.0s; text-align:left; visibility:visible; } [title-text]:after { opacity:1.0; content:attr(title-text); text-align:left; left:50%; background-color:yellow; color:black; font-size:10pt; position:absolute; padding:1px 5px 2px 5px; white-space:pre; border:1px solid red; z-index:1; visibility:hidden; } [title-text] {position: relative;} ::selection{background-color:yellow !important; color:black !important;} ::-moz-selection{background-color:yellow !important; color:black !important;} </style> </head> <body>  <form name="form1" method="post" action="">  <table width="$TableWidth" align="center" border="0" cellspacing="1" cellpadding="3"> <tr><td colspan="99" style="color:white; background-color:#000066; border:2px solid white; border-radius:8px 8px 0px 0px;">$_INTERFACE_TITLE_</td></tr> </table>  <table width="$TableWidth" align="center" border="0" cellspacing="1" cellpadding="3"> <tr><td style="line-height:175%; border-radius:0px 0px 0px 8px;" width='33%'>Acceleration Rate (m/s²)</b><br><input name="a" type="text" value="$a" size="24" maxlength="32"></td> <td style="line-height:175%;" width='33%'>Distance Value in Light Time Units<br><input name="D" type="text" value="$D" size="32" maxlength="81"></td> <td style="line-height:175%; border-radius:0px 0px 8px 0px;" width='33%' title=' TIME UNITS SYMBOLS \n\n S\tSeconds \n M\tMinutes \n H\tHours \n D\tDays \n W\tWeeks \n Y\tYears (365.25 days) \n\n NOT CASE-SENSITIVE '>Time Units Symbol<br><input name="TimeUnitsSymbol" type="text" value="$TimeUnitsSymbol" size="2" maxlength="1"></td></tr> </table>  <table width="$TableWidth" align="center" border="0" cellspacing="1" cellpadding="3"> <tr><td colspan="99" style="background-color:black;"><input type="submit" name="SubmitButton" value=" C O M P U T E "></td></tr>  <tr> <td colspan='3' style='background:transparent; color:black; font-size:10pt; text-align:center;'> <a href='View-Source-Code.php' target='_blank' style='font-family:Verdana; color:black; background:yellow; text-decoration:none; border:1px solid black; padding:4px; font-size:10pt; border-radius:4px;'>  <span style='font-weight:normal;'>View/Copy PHP Source Code</span> </a> </td> </tr> </table>  <table width="$TableWidth" align="center" border="0" cellspacing="1" cellpadding="3"> <tr> <td colspan="99" style="text-align:center; color:GreenYellow; background-color:black;">Double-Click Within Text Area to Select ALL Text<br> <textarea ID="TextArea1" name="TextArea1" style="color:$TxColor; background:$BgColor; padding:6px; border:2px solid white;" cols="$Text1Cols" rows="$Text1Rows" ReadOnly OnDblClick="this.select();" OnMouseUp="return true;"> $TextArea1Text </textarea> </td> </tr> </table>  <table width="$TableWidth" align="center" border="0" cellspacing="1" cellpadding="3"> <tr> <td colspan="99" style="text-align:center; color:GreenYellow; background:black;">Double-Click Within Text Area to Select ALL Text<br> <textarea ID="TextArea2" name="TextArea2" style="color:black; background:white; padding:6px;" cols="$Text2Cols" rows="$Text2Rows" ReadOnly OnDblClick="this.select();" OnMouseUp="return true;"> $TextArea2Text </textarea> </tr> </table>  <table width="$TableWidth" align="center" border="0" cellspacing="1" cellpadding="3"> <tr> <td colspan="99" style="color:GreenYellow; background:black;">PHP Program by $_AUTHOR_<br><span style="color:silver; background:black;">$_REVISION_DATE_</span></td> </tr> </table> </form>   <br><br><br><br><br><br><br><br><br><br><br><br> <br><br><br><br><br><br><br><br><br><br><br><br> </body> </HTML> HTML_WEB_PAGE; /* ############################################################################ This function computes the arbitrary-precision natural logarithm for any positive (x) argument to up to 100 decimals. SPECIAL CASE: If (x) is a decimal value (0 < x < 1), then Ln(x) = -Ln(1/x) ERRORS: Returns error message if non-numeric or if (x <= 0). NO DEPENDENCIES ############################################################################ */ function BC_Ln ($xArgStr, $Decimals=16) { $x = trim($xArgStr); if (StrPos($x, '-') !== FALSE or $x <= 0 or !Is_Numeric($x)) {return "ERROR: BC_Ln()\n\nArgument must be numeric and > 0";} $q = trim($Decimals); $Q = 100; if ($q > $Q) {$q = $Q;} if ($q <= 0) {$q = 1;} // ------------------------------------------------------- // Handle case of fractional (x) values where (0 < x < 1). $sign = ''; if ($x > 0 and $x < 1) {$sign = '-'; $x = bcDiv('1', $x, $Q);} // ------------------------------------------------------------------------------- // Natural logarithm of 10 to 110 decimals (used as internal conversion constant). $LN_10 = '2.30258509299404568401799145468436420760110148862877297603332790096757260967735248023599720508959829834196778403'; /* ------------------------------------------ Determine power of 10 in positive (x) and where to place the decimal in the result. */ $dpos = 1; if($x >= 10) { $dpos = floor(log10($x) + 1); $x = Str_Replace('.', '', $x); $x = substr($x,0,1).'.'.substr($x,1, StrLen($x)); } $PrecisionLimit = '0.'.Str_Repeat('0', $q).'1'; $n = $term = 1; $sum = 0; $k = bcDiv(bcSub($x, '1', $Q), $x, $Q); while (abs($term) > $PrecisionLimit) { $numer = bcpow($k, $n, $Q); $denom = $n; $term = bcDiv($numer, $denom, $Q); $sum = bcAdd($sum, $term, $Q); $n++; } $sum = bcAdd($sum, bcMul($dpos-1, $LN_10, $Q), $Q); // ------------------------------------------------------- // Return the sum rounded to the given number of decimals. return $sign.bcAdd($sum, Str_Replace('1','5', $PrecisionLimit), $q); } // End of BC_Ln(...) function Encomma ($BCNumStr) { $N = trim($BCNumStr); $i = StrPos($N, '.'); if ($i === FALSE) {$N .= '.0';} if (substr($N,0,1) == '.') {$N = "0$N";} $DecimalPart = StrRChr($N, '.'); $i = StrPos($N, '.'); $IntegerPart = Number_Format(substr($N,0,$i)); return "$IntegerPart$DecimalPart"; } function Dist_Accelerated ($AccelRate, $ProperTimeInterval, $Decimals=16) { $a = trim($AccelRate); $t = trim($ProperTimeInterval); $c = '299792458'; $Q = 50; $q = trim($Decimals); $at_c = bcDiv(bcMul($a, $t, $Q), $c, $Q); $c2_a = bcDiv(bcMul($c, $c, $Q), $a, $Q); $X = bcDiv(bcMul($c, $c, $Q), $a, $Q); $Y = bcAdd('1', bcMul($at_c, $at_c, $Q),$Q); $D = bcMul($c2_a, bcSub(bcSqRt($Y, $Q), '1', $Q),$Q); return $D; } /* ########################################################################### This functions returns the formal name of a large numeric value string. ONLY the integer part is used. Any decimal part is ignored. EXAMPLE: Given a signed numerical string argument like: BigNumStr = '7,805,769,880,904,240,738,817,015,808' or BigNumStr = '78057691843657209880904240738817015808' If there is any decimal part of BigNum, it will be removed. Only the integer part counts here. The function returns the formal designation: '7.81 octillion' ########################################################################### */ function BC_Big_Num_Name ($BigNumStr, $Decimals=2) { /* ------------------------------------------------------------- Filter out any commas or negative sign and use absolute value of big number by removing any preceding numerical +/- sign. */ $BigNum = Str_Replace(',', '', Str_Replace(' ' ,'', trim($BigNumStr))); // ------------------------------- // Preserve any numerical +/- sign // to reattach to output. $NumSign = ''; if (substr($BigNum,0,1) == '-') {$NumSign = '-';} if (substr($BigNum,0,1) == '+') {$NumSign = '+';} $BigNum = Str_Replace('+', '', Str_Replace('-', '', $BigNum)); /* --------------------------------- Count number number of raw digits in the filtered numeric string. */ $DigitsCount = StrLen($BigNum); // ------------------------------------ // Error if big integer is non-numeric. if (!Is_Numeric($BigNum)) {return 'ERROR: Cannot resolve as a valid number value.';} /* -------------------------------------------------- Cut off any decimal part of the big number string. Only the big integer part counts here. */ $i = StrPos($BigNum, '.'); if ($i === FALSE) {$BigNum = "$BigNum.0";} $i = StrPos($BigNum, '.'); $BigNum = substr($BigNum, 0, $i); // ----------------------------------------------- // Set number of decimals to use in output string. // This usually just 1 to 3 decimals. $decimals = abs(IntVal(trim($Decimals))); if ($decimals == 0) {$decimals = 1;} /* ---------------------------------------- Define an array of formal integer names. https://en.wikipedia.org/wiki/Names_of_large_numbers */ $DigitGroupsStr = ''; $BigNum = StrRev($BigNum); // *************** // *************** $groups[1] = ''; $groups[2] = 'thousand'; $groups[3] = 'million'; $groups[4] = 'billion'; $groups[5] = 'trillion'; $groups[6] = 'quadrillion'; $groups[7] = 'quintillion'; $groups[8] = 'sextillion'; $groups[9] = 'septillion'; $groups[10] = 'octillion'; $groups[11] = 'nonillion'; $groups[12] = 'decillion'; $groups[13] = 'undecillion'; $groups[14] = 'duodecillion'; $groups[15] = 'tredecillion'; $groups[16] = 'quattuordecillion'; $groups[17] = 'quindecillion'; $groups[18] = 'sexdecillion'; $groups[19] = 'septendecillion'; $groups[20] = 'octodecillion'; $groups[21] = 'novemdecillion'; $groups[22] = 'vigintillion'; $groups[23] = 'unvigintillion'; $groups[24] = 'duovigintillion'; $groups[25] = 'trevigintillion'; $groups[26] = 'quattuorvigintillion'; $groups[27] = 'quinvigintillion'; $groups[28] = 'sexvigintillion'; $groups[29] = 'septenvigintillion'; $groups[30] = 'octovigintillion'; $groups[31] = 'novemvigintillion'; $groups[32] = 'trigintillion'; $groups[33] = 'untrigintillion'; $groups[34] = 'duotrigintillion'; // ---------------------------------------- // Break up big integer into spaced 3-digit // groups and count the number of groups. for($i=0; $i < $DigitsCount; $i++) { $DigitGroupsStr .= substr($BigNum,$i,1); if (($i+1) % 3 == 0) {$DigitGroupsStr .= ' ';} } $DigitGroupsStr = trim(StrRev($DigitGroupsStr)); $DigitGroupsCount = 1 + Substr_Count($DigitGroupsStr, ' '); // ------------------- // Get top group name. $TopNameStr = $groups[$DigitGroupsCount]; // ------------------------------------ // Make number into decimal value (x). $w = StrRev($DigitGroupsStr); $n = StrRev(StrRChr($w, ' ')); $L = StrLen($n); $w = StrRev($w); $w = trim($n).'.'.substr(Str_Replace(' ','',$w), $L-1, StrLen($w)); $w = SPrintF("%1.$decimals" . "f", $w); unset($groups); // Done. return "$NumSign$w $TopNameStr"; } // End of BC_Big_Num_Name (...) /* This function computes the relativistic linear motion statistics Type 1 in the form of a plain text table. Distance are in natural light time units. If time units = years, then distances = Light Years (LY). If time units = days, then distances = Light Days (LD). etc. DEPENDENCY: BC_Ln() ERRORS: No special error checking is done. */ function Relativity_101 ($AccelRateStr='9.80665', $DistStr='1', $TimeUnitStr='Y') { $Q = 60; // Internal working decimals. $q = 50; $c = '299792458'; $a = trim($AccelRateStr); $TU = substr(StrToUpper(trim($TimeUnitStr)),0,1); $D = Str_Replace(',' ,'', trim($DistStr)); /* --------------------------------------------------- Determine which time constant and distance units to use according to the given (TimeUnitsStr). NOT case-sensitive. TIME UNITS and SYMBOLS DISTANCE UNITS and SYMBOLS ---------------------- -------------------------- Seconds S Light Seconds LS Minutes M Light Minutes LM Hours H Light Hours LH Days D Light Days LD Weeks W Light Weeks LW Years Y Light Years LY -------------- -------- ------------------ ------- */ switch ($TU) { case 'S': $K = '1'; $TU = 'Second'; $DU = 'LS'; break; case 'M': $K = '60'; $TU = 'Minute'; $DU = 'LM'; break; case 'H': $K = '3600'; $TU = 'Hour'; $DU = 'LH'; break; case 'D': $K = '86400'; $TU = 'Day'; $DU = 'LD'; break; case 'W': $K = '604800'; $TU = 'Week'; $DU = 'LW'; break; case 'Y': $K = '31557600'; $TU = 'Year'; $DU = 'LY'; break; default : $K = '31557600'; $TU = 'Year'; $DU = 'LY'; break; } // ---------------------------------------------- // Compute relativistic linear motion statistics. $G = RTrim(RTrim(bcDiv($a, '9.80665', $Q), '0'), '.'); $A = bcDiv(bcMul($a,$K, $Q),$c, $Q); $RecipA = bcDiv('1',$A, $Q); $X = bcAdd($D,$RecipA, $Q); $X2 = bcMul($X,$X, $Q); $T = bcSqRt(bcSub($X2,bcMul($RecipA,$RecipA, $Q),$Q),$Q); $beta = bcDiv($T,$X, $Q); $v = bcMul($beta,$c, $Q); $t = bcDiv(BC_Ln(bcMul($A,bcAdd($X,$T, $Q),$Q)),$A, $Q); $TwoT = bcMul('2', $T, $Q); $Twot = bcMul('2', $t, $Q); // ------------------------------------------------------ // Do some extra formatting and speed units computations. $afps = bcDiv($a, '0.3048', $q); $A = bcAdd($A, '0', $q); $T = bcAdd($T, '0', $q); $t = bcAdd($t, '0', $q); $TwoT = bcAdd($TwoT, '0', $q); $Twot = bcAdd($Twot, '0', $q); $vkph = bcDiv(bcMul($v, '3600', $Q), '1000', $q); $vmph = bcDiv($vkph, '1.609344', $q); $v = bcAdd($v, '0', $q); $Ds = (trim($DistStr) <> 1)? 's':''; $Ts = (trim($T) <> 1)? 's':''; $ts = (trim($t) <> 1)? 's':''; $v = encomma($v); $vkph = encomma($vkph); $vmph = encomma($vmph); $beta = bcAdd($beta, '0', $q); // --------------------------------------------------------------------------- // Construct output table in plain, monospace text. $w = " **************************************************************** NUMERICAL RELATIVISTIC SPACECRAFT MOTION TYPE 1 WORKSHEET ############################################################################# APPLIED DATA AND DEFINITIONS: Time Units = $TUs Distance Units = Light $TUs 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 $TUs as the time units, so the value of K = $K = Seconds per $TU. ***************************************************************************** THE BASIC SCENARIO and RELATED NUMERICAL COMPUTATIONS Acceleration rate experienced on the spacecraft. a = $a m/s² = $G G = $afps ft/s² --------------------------------------------------- Distance to the target destination in light units. D = $D Light $TU$Ds ----------------------------------------------------------- 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 = $A --------------------------------------------------- Earth time it takes to accelerate to distance (D). T = SqRt(D² + 2*D/A) = $T $TU$Ts 2T = $TwoT ----------------------------------------------------------------- Proper (spacecraft) time it takes to accelerate to distance (D). t = Ln(A*((D + 1/A) + SqRt(D² + 2*D/A))) / A = $t $TU$ts 2t = $Twot --------------------------------------------- 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 = $beta ----------------------------------------------------------------- Speed achieved at distance (D). This value will always be less than the speed of light (299,792,458 m/s). v = β * c = $v m/s = $vkph km/h = $vmph mi/h ***************************************************************************** "; return RTrim($w); } // End of Relativity_101 (...) // END OF PROGRAM ?>