DETERMINE DUE DISTANCE
Near Earth's surface,
free falling object's velocity increases with time.
Thus, one can describe velocity as a function of time v(t).
v(t) = g × t
free falling object's velocity increases with time.
Thus, one can describe velocity as a function of time v(t).
v(t) = g × t
Simple Calculus
To describe distance traveled by a gforce,
integrate above velocity equation to
describe distance as a function of time.
d(t) =∫v(t) dt = g × t^{2}/2
Above "Newtonian" equations work well for short distances.
Examples of Gforce Distance
However, interstellar travel needs another equation
to describe distance traveled by Thought Experiment's (TE's) notional gforce spaceship.
Perhaps More Calculus .....Recall Einstein, c is always observed as consistent value, c, regardless of the observer's velocity.
Different durations of gforce produce different velocities; thus, observers might observe each other traveling in different speeds, but a few facts stay steadfast:
1. No one ever exceeds c, light speed. 2. All observers consistently measure photons at the same velocity, c= 173.14 AU/day. 3. One day's difference Δ is velocity increase due to one day of gforce acceleration. It's expressed as percent c (or equivalent decimal c).

Previous work leads to following equation to model velocity of gforce spacecraft:
V(t) = (1  (1  Δ)^{t}) c
where:
Following Method could approximate distances without calculus:

A Little Calculus...... 
 ..........Goes a Long Way 

(Many thanks to Wolfram Integrator)
Compute gforce distance: HOW?
Recall basic calculus, the integral, ∫, which is an elegant way of summing the area under the curve under consideration.
Calculus and Motion Problems.
Integration has many practical applications; perhaps the best known is for motion problems. For example, integrating a velocity function is known to yield distance.
For example:
∫_{a}^{b}V(t)dt = d(b)  d(a)
is the general form of a velocity function (with time, t, as the independent variable) being integrated to produce distance traveled from time, a, to time, b.
Most common functions are of the form:
∫k × t^{n }dt = k × t^{n+1}/n+1
EXAMPLE: CONSTANT VELOCITY. Most of us commonly ride automobiles at a constant velocity, such as 60 miles per hour, or cruise in aircraft at 500 nautical miles per hour (knots). Using the latter example:
∫_{0}^{t}k dt = k × t  0
∫_{0}^{2}500 NM/hr dt = 500 knots × 2 hrs  0 = 1,000 NM.
EXAMPLE: CONSTANT ACCELERATION. A more interesting integration might involve gforce acceleration. Let's consider two such examples. First example involves an object free falling from a great height toward the Earth for 10 seconds. Use traditional g=9.8 m/sec^{2}.
∫_{0}^{10 }g × t dt = g t^{2}/2
∫_{0}^{10} 9.8 m/sec^{2} (10 sec)^{2} dt = 9.8 m/sec^{2} (10 sec)^{2}/2  0 = 490 m Second example involves our notional spacecraft gforce accelerating from Earth toward Jupiter for 2.5 days. Use TE's g=0.49 AU/day^{2}.
∫_{0}^{2.5}g × t dt = g t^{2}/2
∫_{0}^{2.5}.5 AU/day^{2} (2.5 day)^{2}dt =.5 AU/day^{2}(2.5 day)^{2}/2 0 = 1.5625AU
Above examples show cases where distances can be easily validated via other computation methods.

However, the case under consideration
( V(t) = (1  (1  Δ)^{t}) c)
needs more sophisticated integration
to determine the corresponding distance function.

INTERPLANETARY RANGES  

After 50 days of gforce acceleration,
methods start to noticeable diverge.  For initial 10 days of gforce acceleration, Newtonian method approximates values determined by the Einsteinian. EXAMPLE: For tenth day, both methods show velocity of just under 5 AUs/day and a total distance of just over 24 AUs. 
BEYOND INTERPLANETARY Short range (10 days) chart contrasts enormously with a longer range chart. After 500 days of gforce acceleration, chart shows greatly diverging values.  

After one year (365 days) of gforce, Newtonian velocity exceeds c. IMPOSSIBLE!!! Thought Experiment's solution shows gforce vessel at 64% c after one year of gforce.


Newton's equation is no good for larger values above chart shows Newton's equation (g × t) exceeding light speed after a year. IMPOSSIBLE!!!  On the other hand, Thought Experiment assumes an Einsteinian version is adequately modeled by an exponential equation which shows at 500 days, a velocity of 131 AUs/day and a total distance of about 40,000 AUs (≈ 2/3 LY). 

VOLUME 0: ELEVATIONAL 

VOLUME I: ASTEROIDAL 
VOLUME II: INTERPLANETARY 
VOLUME III: INTERSTELLAR 
0 Comments:
Post a Comment
Links to this post:
Create a Link
<< Home