### Re: How long at 5g acceleration would it take to reach 99%light speed? 3g?

Date: Wed May 24 12:35:06 2000
Posted By: Sidney Chivers, , Nuclear Engineering, retired
Area of science: Physics
ID: 959050492.Ph
Message:
```
The answer is about 2 years at 5g and about 3 years at 3g.  Now for the
how I got that part.

Force is equal to the change in momentum with respect to time, or

F = dp/dt,
where dp/dt represents the differential change in
p, momentum, with respect to time

or,

change in time in going from rest to 0.99 c (c being speed of light)

t = change in momentum divided by the force,

the force being a constant (mrest) x (a), where
mrest is the mass of the object at rest
and 'a' is the acceleration of the object.

At relativistic speeds, the relativistic mass is considerably larger than
the rest mass, or

mass = mrest divided by the square root of (1 - beta squared), where
beta is the ratio of the object's speed and the speed of light (v/c)

The velocity is just 0.99 c, or the relativistic momentum is

p = [ mrest (beta) c ] divided by the sq. root of (1 - beta squared)

If beta is 0.99, 1-beta is 0.01 and the square root of same is 0.1, or

p = mrest (0.99) c / 0.1  = 9.9 mrest c

Finally,

time = change in momentum / force

= [ 9.9 mrest c ] / [ mrest  a ]  = 9.9 c / a

That's the time elapsed in accelerating from rest to 0.99 of the speed of
light is simply 9.9 times the speed of light divided by the acceleration
rate.

For c = 3 x e 8 meters per second and
g = 9.8 meters / second second   [ or meters per second squared ],

and knowing that 1 year equals 3.156 e 7 seconds

at 3g it takes about 3 years to reach 0.99 c and
at 5g it takes about 2 years to reach 0.99 c.

References:

Taylor and Wheeler, Spacetime Physics, W. H. Freeman and Company, 1966.

Ray Skinner, Relativity for Scientists and Engineers, Dover Publications,
1982.

```

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