How long do you have to accelerate at G to reach the speed of light?

25 May 2013


Dear Dr Chris.

I have plucked up the courage to write to you with a question from a colleague of mine which I can't answer.

I have no idea why he wants to know this ( I think he reads too many science fiction comics or watches too much 'Star Treck' ) but here goes :-

If an object of negligible size & mass is launched from a standing start in a vacuum, and is subjected to an acceleration force of 1 G - how long will it take to reach the speed of light.

Hope you are able to provide an answer, or even a formula to calculate an approximate result when and if you have the time.

Best Regards

Jack Stott BSc(Hon) Elec Eng Science


i found this question, due to myself wondering the same.

forget for a moment that it cant happen.

can somone spell out how fast, with cooresponding time reference, a mythical spaceship would travel, if it started from a standing stop, and accelerated at 1 g constant...

i have no clue on the math, but won't it be something to the effect of 1mph at 1 second, 2 mph at 2 seconds, 8mph at 3 seconds, 32mph at 4 seconds, 128mph at 5 seconds, 512mph at 6 seconds, etc.

i know my numbers are not close to being right, but its simply to illistrate what im asking.

amy ideas?

Assuming no air resistance and Newtonian physics only (no effect from General Relativity), it would take 7,818 seconds or 2.2 hours for an object to achieve light speed if an object is accelerated at the acceleration of Earth gravity from rest (from the point of view of the launch pad).

Here's my math:

I wrote the above post, but I realized I made an important mistake. I've corrected my spreadsheet linked there. Sorry all!

Regarding my question on how long the passenger would have to endure the journey while accelerating to 99.999% of the speed of light at 1G, as opposed to an observer's time frame from a planetary frame of reference, here is the article on the Photon Rocket drive, which will make acceleration to 99.999% of the speed of light possible:

Before I found this site I had already discovered that it would take about 1 year to reach 99.999% of the speed of light (the speed that a Photon Rocket drive could reach) but I could not find anywhere the answer to the question of how long this would be within the frame of reference of the space-ship passengers.

I would very much appreciate it if anyone could answer this question, which I know is a much harder question to answer than the first (I suspect). How long would it take from within the passenger's frame of reference to reach 99.999% of the speed of light at a constant acceleration resulting in 1G? Then I would know that it would take twice as long, including an equal length of time to decelerate, plus a small amount of extra time at very, very close to light-speed (how long exactly for say every 10 light years?) to travel to another star tens of light years away.

This is a very practical question to me as we will soon have much increased life spans and we will hopefully live to see the Photon Rocket become a reality and perhaps even to travel to another star to colonise another planet in another solar system.

An object with non-zero mass (even negligible mass is non-zero) will never reach the speed of light. Due to relativistic effects, each "unit" of acceleration becomes less effective at increasing your velocity (relative to some other object, of course) as your relative velocity approaches the speed of light.

Never? I think you misread the question, but thank you for bringing us up to speed on relativity.

Come on Guys. I feel like I'm living in the Matrix. 1G has to do with the density of the earth. 1Y is totally unrelated and only related to the distance from the sun. So a perfectly formed earth with elements that support life at a perfect distance from the sun with life living in a 1G field that gets you to the speed of light after one year IS NUT!!!! HOW CAN THIS BE!!!

1G is 9.8m/s/s. It is a measure of acceleration. When you drop something on earth, it accelerates towards the ground at 9.8 meters per second per second (or 9.8 meters per second squared). So after the first second, it will be traveling at 9.8 meters per second, after the next second it will be traveling at 19.6 meters per second. If you dropped it from a high enough place the air friction (force pushing up) would equal 1G and the object would stop accelerating. In space, there is no air friction so a space ship could theoretically continue to accelerate at 1G until it reached the speed of light (or the laws of relativity take over and time slows down and you never actually reach light speed....)

The math is fairly simple, but my question is
Why? I assume Einsteins and Newtons equations are interrelated. It can not be coincidence that accelerating at 1g for 1 year equals the speed of light.

It may be physics 101 but that's the wrong formula. That formula would be for constant velocity. The question was about constant
Acceleration which is meters per second squared or an increase of meters per second every second. So after the first second you going 10 meters per second after the second you're going 20 meters per second the third you're going 30 meters per second.

Well, id say if you assume that you can reach the speed of light by normal means and can achieve exactly 1g throughout the trip (I will use 300,000,000 m/s and 10m/s per second as rough approximations for this and assume this is from not moving at all) it would take approximately 30,000,000 seconds (500,000 minutes, 8333.33333333 hours, 347.222222222 days, 49.6031746032 weeks or 0.9539072039 years).

Thanks for the suggested answer, but the original poster did ask for the formula / workings...

its phys 101. Using only newtonian physics, v = v(0)+ gt; and v(0)=0
if we starts from rest:
t = v/g = 300,000,000/10 = 30,000,000s

Not quite right. The formula you provided assumes v = d / t, where t is one unit of time. Thus, your example holds if t (i.e. one unit of time) is 30,0000,000 seconds. However, the v in the formula (d / t), assumes t is one second. There's the disconnect.

Said differently, you need to account for the velocity the object is traveling each incremental second, and add that value to the formula you provided until you get to light speed. Alternatively, you can use the Summation Formula, which involves a simple quadratic equation, to calculate the time in seconds you'd need to achieve light speed.

Here's my math:

I wrote that post, and I was wrong. I've corrected my spreadsheet linked there. Sorry all!

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