Swiss Micros DM42: Time Dilation
Introduction
The program DILATE calculates the time passage from the traveler's perspective:
t' = t * √(1 - (v/c)^2)
where c = Speed of Light = 299,792,458 m/s
We can also state the equation as:
t' = t * cos(arcsin (v/c))
because cos(arcsin(x)) = √(1 - x^2) for |x| ≤ 1
Why cos(arcsin(x)) = √(1 - x^2)?
Let Θ = arcsin x. Then x = sin Θ. Assume that -1 ≤ x ≤ 1.
From the trigonometric identity:
cos^2 Θ + sin^2 Θ = 1
(cos Θ)^2 + (sin Θ)^2 = 1
With sin Θ = x,
(cos Θ)^2 + x^2 = 1
(cos Θ)^2 = 1 - x^2
cos Θ = √(1 - x^2)
cos(arcsin x) = √(1 - x^2)
Swiss Micros DM42 Program: DILATE
Also: Free42, Plus42
00 {52-Byte Prgm}
01 LBL "DILATE"
02 "TIME?"
03 PROMPT
04 "VEL (M/SEC)?"
05 PROMPT
06 299792458
07 ÷
08 ASIN
09 COS
10 ×
11 "T'= "
12 ARCL ST X
13 AVIEW
14 RTN
15 END
Example
A spaceship traveling at a velocity of 186,000,000 m/s (about 416,070,150 mph) for 1 year from our perspective.
XEQ DILATE
TIME? 1
VEL (M/SEC)? 186000000
Result: 0.78426 year would have passed on the spaceship, which is about 9 months and almost 13 days
Until next time,
Eddie
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