### ACCELERATE FOR A YEAR.

Does this apply when object's speed increases beyond c, speed of light?

Einstein, says the speed of light, c, is absolute; thus, no object can attain a speed greater than c. So, these two giants of science appear to disagree.

*(NOTE: Thought Experiment (TE) shows agreement in subsequent text.)*Einstein also says that all observers measure same value for c regardless of observer's own speed. This also seems to conflict with Newtonian concept of speed differential which leads us to believe that faster observers would observe a slower c than slower observers.

To resolve this paradox, TE suggests the remainder concept which resembles the complement concept of Probability Theory.

See associated table for tabulated results..

BACKGROUND:Thought experiment further assumes technology advances sufficient to produce a well controlled, consistent flow of exhaust particles at a set speed. Such exhaust particle flows would provide sufficient momentum such that a relatively small mass of fuel could accelerate a larger spaceship at g-force to simulate gravity and gain enormous speeds. Such speeds would require spaceship to g-force decelerate for same duration as acceleration. QUESTION: How does Probability Theory's"Complement" Concept relate to interstellar g-force velocities????COMPLEMENT: For any spin of the Roulette Wheel (1st spin, 2nd spin, 100th spin, etc.), the odds of spinning 00 is always 2.6% and the complement (i.e. "odds of not spinning 00") is always 97.4%.LIGHT SPEED REMAINDER ANALOG: For any day of g-force acceleration (1st day, 2nd day, 100th day, etc.), vessel always attains velocity of .283%c; remainder (remaining velocity til c, light speed) is always 99.717% c. | To illustrate probability, consider an American roulette wheel which has: **18 red slots****18 black slots****two green slots (0 and 00)**
P(00 ¦ 1 spin) = 1/38 = .026COMPLEMENT: If the event of interest is spinning a 00; then, the event's complement is spinning any other slot (i.e., not spinning a 00).To compute the probability of NOT rolling a double zero on first roll, use the complement concept as shown: P(~00¦1 spin)=37/38=.974=1 - P(00¦1 spin) | |||||||||||||||||||||||||||||||||||||||||||||||

P(x) is the probability of event x occurring; P(~x) is the probability of x not occurring (complement of x):P(~x) = 1 – P(x)EXAMPLE: Use the complement's probability to determine probability of rolling at least one double zero (00) in three rolls of the wheel? First, determine the probability of a double zero on the first roll. This is easily calculated by dividing number of possible occurrences by total number of possible events. P(00 ¦ 1 spin) = 1/38 = .026
Next, use the complement to determine probability of NOT rolling a double zero on first roll:
P(~00 ¦ 1 roll) = 37/38 = .974 = 1 - P(00 ¦ 1 roll)
Use that value to determine probability of NOT rolling 00 during three successive rolls:
P(~00 ¦3 rolls)=P(~00 ¦1 spin)×P(~00 ¦1 spin)×P(~00 ¦1 spin)P(~00 ¦ 3 spins) = P(~00 ¦ 1 spin)^{3 }= .924
Finally, use the complement once again to determine likelihood of at least one double zero during three rolls of the American roulette wheel.
P(00 ¦ 3 spins) = 1 - P(~00 ¦ 1 spin)^{3 }= .076
What about 10 or any number (n) of spins?? What are the chances of getting at least one double zero?
P(00 ¦ 10 spins) = 1 - .974^{10} = 1 - .768 = .332P(00 ¦ n spins) = 1 - P(~00 ¦ 1 spin) ^{n} |
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Daily Increase in Velocity. After first day of g-force acceleration, velocity can be determined as follows:
Daily Difference. Velocity after first day's g-force acceleration can be expressed as a portion of light speed, c ( = 299,792 km/sec). TE calls this the "daily difference". Δ = .00283 c = .283% c |
With exponentials, the complement concept turns a very difficult probability calculation into a very simple one. Note following points:
- For any particular roll, the odds of rolling a double zero remain 1/38 regardless of how many previous rolls were not double zero.
- Though initial roll's complement is quite high (97.4%), prediction of repeated rolls will continue to whittle down the probability of NOT rolling a 00.
- For an objective predictor who calculates probabilities prior to any gaming, they notice that as number of rolls increase, probability of 00 increases.
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For interplanetary destinations,g-force velocitiesclosely approximate linear Newtonian values. | ||||||||||||||||||||||||||||||||||||||||||||||||
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Ship departs Earth and accelerates at g-force. After precisely one day of g-force flight, ship ejects Module-1 with observer. Since Module-1 has no propulsion system, it will maintain constant speed, v_{1}, while g-force ship will continue increasing speed. Earth bound observer measures c at 299,792,458 m/sec and measures Module One's velocity at .283%c. NOTE: Remaining velocity till light speed ("Remainder") is 99.717%c. | ||||||||||||||||||||||||||||||||||||||||||||||||

Second Day Scenario: After precisely two days of g-force flight, ship ejects Module Two (M2) with another observer. M1 observes M2 at .283%c; thus, a remainder of 99.717%c. Earth observes M2 at .566%c; thus, a remainder of 99.434%c. However, all observers measure same value for c, regardless of platform velocity. (Recall Einstein)
In fact, all observers (Earth, M1, M2, g-force ship) are at different speeds; yet, they all observe c at same value: 299,792,458 m/sec.
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This is true during start, middle or end of flight.However, Earth bound observer will measure different velocities for different modules. First few modules appear to increase speed at a linear rate. | ||||||||||||||||||||||||||||||||||||||||||||||||

Succeeding Day's Velocities can also be expressed as percentages of light speed (%c). Using Newton's linear method, we get values in table. Eventually, this violates Einstein's Special Relativity Theory by increasing speed above c. This paradox can be restated:G-force velocities approximate a linear relationship for short durations, but this linearity cannot be sustained. How can we resolve this paradox?? | PROBLEM!!! Linear model predicts g-force vehicle will exceed c. Einstein says this can't happen. Most scientists agree; thus, TE considers another way to compute g-force velocities. | |||||||||||||||||||||||||||||||||||||||||||||||

For interstellar destinations, g-force velocities must use ingenious, Einsteinian values. | ||||||||||||||||||||||||||||||||||||||||||||||||

To account for relativity, use remainders and exponentials to compute interstellar velocities. | ||||||||||||||||||||||||||||||||||||||||||||||||

Daily Difference (Δ)
| Remainder (R)
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Exponentials.
| Daily Speeds.
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| First few days of g-force demonstrate approximate agreementbetween Newtonian and Einsteinian models. | |||||||||||||||||||||||||||||||||||||||||||||||

Day 200 Scenario.Like all the previously ejected modules, M-199 observes velocity of the subsequent module, M-200, at .283%c with 99.717%c remaining until light speed. | ||||||||||||||||||||||||||||||||||||||||||||||||

The earth bound monitor observes all 200 modules.
Subsequent modules attain faster and faster velocities
as they're ejected throughout the voyage.
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| For greater durations,values diverge even more betweenthe models of Newton and Einstein. | |||||||||||||||||||||||||||||||||||||||||||||||

In fact, we eventually see impossible values for Newtonian method.After 354 days, Newtonian method exceeds c, an impossible feat.
Thought experiment assumes Einsteinian method more accurately determines g-force velocities for longer durations. | ||||||||||||||||||||||||||||||||||||||||||||||||

SUMMARY | ||||||||||||||||||||||||||||||||||||||||||||||||

Newtonian Method closely approximates velocities for first few days of g-force flight.Δ (Daily Difference of g-force velocity) remains a constant 0.489 AU/day ≈ 0.283% c. For convenience, thought experiment assumes following formula:.489 AU/day × t _{Day} = V_{Fin} = Δ× t _{Day }= .283%c× t_{Day}
works well for short interplanetary flights. Thus, 10 day interplanetary trip would reach velocity of 4.89 AU/day ≈ 3% c, well short of light speed. Relativity concerns prevent this linear increase to go on indefinitely, but it's a fairly good model for interplanetary travel with g-force propulsion systems. For much longer interstellar voyages, this formula will need adjustment to produce usable values.
Newtonian values don't prove true for much greater durations. (Example: After 354 days of g-force, above formula predicts velocity exceeds c, an impossibility.)
| Einsteinian Methodmore closely approximates g-force speeds for interstellar flights.
Thus, TE assumes following equation:
V_{t} = c - c(1 - Δ)^{t}G-force interstellar flight durations take years (much longer than the few days required for interplanetary flights). The distances are too vast and the flight durations are too great; thus, we cannot get by with the convenient heuristic used for Volume 2, Interplanetary.Following cells demonstrate how we can leverage a simple probability concept to derive above equation. | |||||||||||||||||||||||||||||||||||||||||||||||

Spin the Wheel | G-force to the Stars. | |||||||||||||||||||||||||||||||||||||||||||||||

Probability of rolling a double zero on first roll is very low. | G-force velocity after first day of flight is very low, a small percentage of c. | |||||||||||||||||||||||||||||||||||||||||||||||

P (00) = 1/38 = .026 = 2.6% | V (1 day g-force) = 847 km/sec ÷ 299,792 km/sec = .283% c = V_{1dy} | |||||||||||||||||||||||||||||||||||||||||||||||

Complement(00): P (~00) =1 - P(00) = 97.4% = C_{1Spin} | Remainder: c - V_{1dy} = 99.717% = R_{1dy} | |||||||||||||||||||||||||||||||||||||||||||||||

If one knows the probability of an event's complement; then, easily determine the probability of the event: P(E) = 1 - P(C) | If one knows the remainder of an g-force vessel's velocity; then, one knows the velocity of the vessel:V_{t} = c - R^{t} | |||||||||||||||||||||||||||||||||||||||||||||||

Spin 38 Roulette Wheels Intuitively, it seems that at least one must show double zero. NOT SO!!!! | After accelerating for one year (365.25 days of g-force), Newtonian physics leads one to believe that velocity will exceed c, light speed. NOT SO!!! | |||||||||||||||||||||||||||||||||||||||||||||||

After the first few spins of the wheel, it looks like the probabilities are additive. But after many spins, this proves to not be true. | After the first few days of g-force, the resultant velocities appear to add. However, Einsteinian Relativity demonstrate that g-force velocities cannot continue to add indefinitely. | |||||||||||||||||||||||||||||||||||||||||||||||

To observe results of 38 spins, one could record each spin in sequence. Alternatively, 38 observers could simultaneously spin 38 separate wheels and record the results. | To observe one year of g-force velocities, an Earth bound observer can measure all 365 daily velocities of a g-force vessel. Alternatively, one could conveniently observe 365 markers with constant velocity achieved at time of ejection. | |||||||||||||||||||||||||||||||||||||||||||||||

As number of spins increase, complement of double zero decreases; this is easily computed with exponentials.C(00| n Spins) = .974^{n}= C_{n-Spins} | As g-force flight duration increases, decreasing light speed remainder is easily computed with exponentials. EXAMPLE: After 1 year of g-force, Remainder (R) decreases to .99717^{365.25}= .3552.V_{1Year }= c - R_{365.25days }= c (1 - .99717^{365.25}) = 64.48%c | |||||||||||||||||||||||||||||||||||||||||||||||

The complement of the complement gives the probability that at least one double zero will happen during a given quantity (n) of spins. | The remainder of the remainder gives us the velocity of the vessel after a given quantity (t) of days with g-force propulsion. See associated 1G TABLE: Accelerate for 1 Year. | |||||||||||||||||||||||||||||||||||||||||||||||

P(00| n Spins) = 1 - C_{n-Spins} | V_{t} = c - R_{t}_{ }= c - c(1 - Δ)^{t} |

VOLUME 0: ELEVATIONAL |
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VOLUME I: ASTEROIDAL |

VOLUME II: INTERPLANETARY |

VOLUME III: INTERSTELLAR |

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