- #1
Star Drive
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This solution is from my notes about twenty years ago and the integration by form was made by a friend. We worked on this interesting question together. Many others have surely gone down this same road. I am not a physicist, nor have I had University Physics since 1968.
It should be realized that for a constant applied force F, the acceleration magnitude will taper off along a steep curve, as the relativistic effects increase the mass of the object being accelerated. As the object approaches the speed of light in the limit, the acceleration tends to fall to zero in the limit. In such an extreme case, the object would become infinitely massive in the limit, as it approaches light speed. This is a key reason why the speed of light can’t be exceeded.
c =Velocity of Light, assumed constant at 2.998 x E8 m/s.
t =Time in Seconds.
\Pi= 3.141593
(Approximately)
Capital letters represent the initial or rest quantities as viewed by the observer. In example; A = Initial Acceleration, M = Initial Mass., F = Initial Force (Held constant throughout mission)
Lower case Letters represent the instantaneous variables, as viewed by the observer. In example; a = Instantaneous acceleration, m = Instantaneous mass.
Fundamental physics relations apply;
a=dv/dt
F=ma
Equivalent relativistic factors [T] follow;
[T]=1/\sqrt{1-v^2/c^2}
[T]=c/\sqrt{(-v)^2 + c^2}
Instantaneous acceleration;
a=dv/dt
Instantaneous (relativistic) mass;
m=M[T]
The following basic substitutions yield;
F=ma
F=M[T]dv/dt
Fdt =M[T]dv
(F/M)dt =[T]dv
\int(F/M)dt = \int[T]dv
Where; F, M and c are constants
(F/M) \int dt= c \int dv/ \sqrt{(-v)^2+c^2}
Integrate Equation by form with CRC integral tables to obtain the following result.
(F/M) t=[c/\sqrt{1}] [\arcsin(v\sqrt{1/c^2})]
Solve relationship F=MA for A and substitute. (All initial Values)
At=c[\arcsin(v/c)]
At/c=\arcsin(v/c)
\sin[At/c]=v/c
The following equation is the primary relativistic solution for velocity, as a function of time, where the units are angles (radians) and time (seconds).
v =c[\sin(At/c)]
Note: The following constraints apply;
[At/c]<(\Pi/2)
t <(1/A) [(\Pi)(c)/2)]
t< (1/A) [4.7092] (10^8)
The following equation is the derived solution for time to any arbitrary speed, which is less than light speed.
t =(c/A)[\arcsin(v/c)]
As an example solution, for 1.0 Earth gravity initial acceleration, the time to 99.9% of the velocity of light is about 1.4788 years. (The initial force F is held constant)
It was surprising for the solution to be inversely linear. In example, it requires about 14.788 Years for the same results at 0.1 G.
A personal computer spreadsheet was used to iterate the solution for time at some 10,000 points along the curve from rest to 99.9% of the velocity of light. That result agreed very well for the mathematical solutions provided above, with t = 1.48 years at 1.0G. This machine computation is correct since F=ma and v=at is true, for very small changes of velocity v.
By summing the resulting time, for each of the 10,000 points with the PC spreadsheet, it was possible to approximate the integration for time. Please do have some fun and setup a spread-sheet for your self.
DISCUSSION POINTS:
It seems that the math yields a false solution for c >= speed of light. As the math implies, does the spacecraft velocity actually ‘slow down’, with continued acceleration, ‘after’ the speed of light is ‘exceeded’? Discussion: What does this really mean? LOL
Is anyone else surprised by the appearance of trigonometric functions in the two solutions? Discussion?
Is anyone else surprised by the apparent inverse linearity of the solution for time with respect to acceleration? Discussion?
As a thought experiment, consider a one-way Interstellar probe, which must be robotic and maintenance free The probe’s initial mass should be 100 metric tons. The chosen power plant must provide an initial or starting acceleration of 0.1 Earth gravities. The power plant must provide a constant reaction force F to propel the vehicle and operate continuously over the entire mission. The probe must accelerate from rest, relative to the mission planner, to a maximum velocity of 99.5% of the speed of light. Such a velocity would cause a relativistic mass increase of about x10 and a time compression of about x(1/10). Would the rocket engineers comment on the practical aspects of building such a vessel? Possible or impossible? Discussion?
It should be realized that for a constant applied force F, the acceleration magnitude will taper off along a steep curve, as the relativistic effects increase the mass of the object being accelerated. As the object approaches the speed of light in the limit, the acceleration tends to fall to zero in the limit. In such an extreme case, the object would become infinitely massive in the limit, as it approaches light speed. This is a key reason why the speed of light can’t be exceeded.
c =Velocity of Light, assumed constant at 2.998 x E8 m/s.
t =Time in Seconds.
\Pi= 3.141593
(Approximately)
Capital letters represent the initial or rest quantities as viewed by the observer. In example; A = Initial Acceleration, M = Initial Mass., F = Initial Force (Held constant throughout mission)
Lower case Letters represent the instantaneous variables, as viewed by the observer. In example; a = Instantaneous acceleration, m = Instantaneous mass.
Fundamental physics relations apply;
a=dv/dt
F=ma
Equivalent relativistic factors [T] follow;
[T]=1/\sqrt{1-v^2/c^2}
[T]=c/\sqrt{(-v)^2 + c^2}
Instantaneous acceleration;
a=dv/dt
Instantaneous (relativistic) mass;
m=M[T]
The following basic substitutions yield;
F=ma
F=M[T]dv/dt
Fdt =M[T]dv
(F/M)dt =[T]dv
\int(F/M)dt = \int[T]dv
Where; F, M and c are constants
(F/M) \int dt= c \int dv/ \sqrt{(-v)^2+c^2}
Integrate Equation by form with CRC integral tables to obtain the following result.
(F/M) t=[c/\sqrt{1}] [\arcsin(v\sqrt{1/c^2})]
Solve relationship F=MA for A and substitute. (All initial Values)
At=c[\arcsin(v/c)]
At/c=\arcsin(v/c)
\sin[At/c]=v/c
The following equation is the primary relativistic solution for velocity, as a function of time, where the units are angles (radians) and time (seconds).
v =c[\sin(At/c)]
Note: The following constraints apply;
[At/c]<(\Pi/2)
t <(1/A) [(\Pi)(c)/2)]
t< (1/A) [4.7092] (10^8)
The following equation is the derived solution for time to any arbitrary speed, which is less than light speed.
t =(c/A)[\arcsin(v/c)]
As an example solution, for 1.0 Earth gravity initial acceleration, the time to 99.9% of the velocity of light is about 1.4788 years. (The initial force F is held constant)
It was surprising for the solution to be inversely linear. In example, it requires about 14.788 Years for the same results at 0.1 G.
A personal computer spreadsheet was used to iterate the solution for time at some 10,000 points along the curve from rest to 99.9% of the velocity of light. That result agreed very well for the mathematical solutions provided above, with t = 1.48 years at 1.0G. This machine computation is correct since F=ma and v=at is true, for very small changes of velocity v.
By summing the resulting time, for each of the 10,000 points with the PC spreadsheet, it was possible to approximate the integration for time. Please do have some fun and setup a spread-sheet for your self.
DISCUSSION POINTS:
It seems that the math yields a false solution for c >= speed of light. As the math implies, does the spacecraft velocity actually ‘slow down’, with continued acceleration, ‘after’ the speed of light is ‘exceeded’? Discussion: What does this really mean? LOL
Is anyone else surprised by the appearance of trigonometric functions in the two solutions? Discussion?
Is anyone else surprised by the apparent inverse linearity of the solution for time with respect to acceleration? Discussion?
As a thought experiment, consider a one-way Interstellar probe, which must be robotic and maintenance free The probe’s initial mass should be 100 metric tons. The chosen power plant must provide an initial or starting acceleration of 0.1 Earth gravities. The power plant must provide a constant reaction force F to propel the vehicle and operate continuously over the entire mission. The probe must accelerate from rest, relative to the mission planner, to a maximum velocity of 99.5% of the speed of light. Such a velocity would cause a relativistic mass increase of about x10 and a time compression of about x(1/10). Would the rocket engineers comment on the practical aspects of building such a vessel? Possible or impossible? Discussion?