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


SO - there appeared to be some confusion between what guess is, and what a theory is:
From the Dictonary:

noun: theory; plural noun: theories
a) a supposition or a system of ideas intended to explain something, especially one based on general principles independent of the thing to be explained.
"Darwin's theory of evolution"

b) a set of principles on which the practice of an activity is based.
"a theory of education"

c) an idea used to account for a situation or justify a course of action.
"my theory would be that the place has been seriously mismanaged"

a) a collection of propositions to illustrate the principles of a subject. <--- (physics is essentially applied math - as in we use math so this would be the definition that fits best in this case)

Here's a different definition of a theory put very simply:
A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment. Such fact-supported theories are not "guesses" but reliable accounts of the real world.

or this one:

scientific theory
a coherent group of propositions formulated to explain a group of facts or phenomena in the natural world and repeatedly confirmed through experiment or observation:

Or you can just do a google search for the phrase 'scientific theory' if you STILL think a scientific theory is akin to an educated guess and you will be hard pressed to fine anything that substantiates your thinking.

"The way that scientists use the word 'theory' is a little different than how it is commonly used in the lay public," said Jaime Tanner, a professor of biology at Marlboro College. "Most people use the word 'theory' to mean an idea or hunch that someone has, but in science the word 'theory' refers to the way that we interpret facts."

"Hypothesis. Theory. Law. These scientific words get bandied about regularly, yet the general public usually gets their meaning wrong."

Please, strive to be better than the general public.

I did a bit of dictionary work myself...


Definition: excessively concerned with minor details or rules; overscrupulous.

Hmm, the cap seems to fit.

It would take less than 12 parsecs.

A parsec is an astronomical unit of length not a measure of time.

Einstein in his theory of relativity said a clock will slow down in a stronger gravity field . So velocity it self will not slow time.for example voyager 2 is traveling at 44000 miles an hour and it’s clock is 35seconds faster than the one a nasa . It slowed down when it left earth again each time it made a sling shot around a planet now it is in free minuscule gravity it’s clock is counting seconds faster

If your rest mass m is 0.5 eV, or the same rest mass as a neutrino, it takes no time at all, at any value of acceleration.

Larger masses are larger amounts of bound energy, and can NEVER reach the speed of light, at any acceleration. Acceleration stops when the remaining amount of acceleration would require the energy of the particle or aggregation of particles to exceed the speed of light minus the rest equivalent of their Compton wavelength energies, which limits everything made of matter to velocities less than c. The neutrino may be the only exception to the rule, because of the way its internal inertia / energy is configured.

No, you won't find this in your physics or spacetime-based relativity textbooks.

Yes. This is in the books of physics. It considers dilatation of time, length and mass.
You are right about any mass, including the mass of the neutrino. But not for the vague reasons you put there. Only a photon travels at the speed of light. Nothing else does.
If you cannot find it in your physics or spacetime based relativity textbook, try the ones written for the university, especially for the courses majoring in physics or astrophysics. You can even find equations to calculate exactly what is going on.

The human mind is not capable of understanding the vastness and infinite universe and all that it contains . The largest number , calculated by the most powerful computer , is how many times bigger and larger and greater the universe is than believed by modern scientists . The big bang ? From which universe ? One big bang , or one billion big bangs occurring simultaneously ? Humans cannot understand infinity.

Why can't you hypothetically reach light speed using an interstellar ramjet? No propellant to accelerate, yes I know mass increases but only significantly near C. o go 90% C.

...then why is the universe still expanding AND accelerating?
I mean relativity and basic math indicate the universe should have hit the so-called "Speed-Limit" of (C) a long time ago... So what gives?

The expansion of the universe is the expansion of spacetime itself. The speed of light "speed limit" only applies to objects travelling through spacetime.
Indeed you are correct about the expansion exceeding C : for regions of the universe that are extremely far away, the distance is increasing at a pace faster than light due to the expansion of the space in between.

Right off the bat, you will need to use an equation that relates acceleration and time. In this case since the object's initial velocity, final velocity, and acceleration is given we use the formula: a=Δv/Δt. ( speed of light is 3x10^(8) and acceleration of gravity or G is 9.8m/s^2. a=(vf-vi)/t , a=(3x10^(8)-0)/t , 9.8t=3x10^(8), t=30612244.9 seconds or 354.31 days. Now since we put ourselves in the perspective of the spaceship, assuming that it was accelerating at 9.8m/s^2 and that it will eventually reach the speed of light. Now that is the time observed from the spaceship, what if you were on earth so how much time would have passed ? Well, now you can apply einstien's theory of relativity. The formula for which is, (time observed from earth) Te = Ts (time observed from spaceship) / Sqrt* 1 - (velocity of the object)v^2 /c^2 (speed of light). For this equation however, we cannot reach the speed of light, and approximate by using 99% of the speed of light. Te=30612244.9/Sqrt*1-0.99, Te=30612244.9/0.1, Te=306122449 seconds or approx. 9.71 years as observed from earth.

You forgot to square the .99

Current tech of our primitive so called Warp tech will allow a large cargo/colony ship to safely travel at around 10% light speed with a safe acceleration of around 10% 1G and would take about 1 year to reach that speed, during acceleration there would be G forces felt........ as the ship and all contents would be in a sense FALLING toward its destination. Don't forget that you will also need about 1 year to slow down projecting a gravity well behind the ship. Alpha Centauri is nearest and about 4 light years away, so it would take about 50 years to reach it.With the current acceleration of our technology this travel time could be reduced to more than half in the next 15 years, thus another ship would reach Alpha Centauri before the first one did........

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?


Yes, you understand correctly the concept. Acceleration due to gravity (on Earth) is 9.8 m/s or 32 feet per second/per second. 32fps = 21.8 miles per hour. So at the end of the first second, the velocity V will be 21.8 mph. After 2 seconds, V= 21.8 x 2 = 43.6 mph and so on. The speed of light in miles per hour = 6.706 x 10^8 mph = 67,060,000,000 mph or around 67 million miles per hour. From these numbers, I calculated almost 10 years to reach 10% the speed of light.

You say "the velocity V will be 21.8 mph. After 2 seconds, V= 21.8 x 2 = 43.6 mph and so on." but is the third value additive or exponential? I.e. is the 3 seconds number 65.4 mph (21.8+21.8+21.8) or 87.2 mph (21.8x2x2).

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!

You're right. More precicely about 355 days. The math is simple conversion using 186,000 miles per second for the speed of light and 32 ft/sec/sec for 1 gravity acceleration.

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:

After travelling say 50 light years. How much time would have passed on earth? On the craft a year to reach light speed 48 years at light speed -1 then a year to slow to a stop. OR would the 50 years pass on earth and only 2 year+ pass on the craft?

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.

This website will calculate the ship time and observer time.

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.

I think we should quit assuming that Einstein's theories were correct.A theory is just a mathematical guess

I really want to slap people when they say something like... "This is just an educated guess" that doesn't even meet the standard for hypothesis. A theory isn't any sort of guess. It has a body of evidence experimentation and refinenment... peer reviews, challenges etc. Hypothesis is a well reasoned POTENTIAL explanation to an observable phenomenon. That hypothesis isn't a guess. It is based off the body of evidence that has come before. So, when someone says that about a theory, the first thing I think is "Flunked general science."

yeah but it’s still a guess.. wrap it how you want.

Theory is not a guess. That's not what a theory is.

The theory of relativity is as well proven as the theory of gravity or germ theory. We have to compensate for the effects of relativity in our satellite communications/tracking equipment.

NOBODY is just "assuming" Einstein's theories are right. That's not how science works either.

Every grade school science related book starts chapter 1 by explaining what a
Hypothesis and theory is.

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....)

So what if our units of acceleration and time were defined by some earth-specific quantities? Acceleration and time themselves are not specific to earth. A metre might originally have been defined as one ten-millionth of the distance from the Earth's equator to the pole, but we can still use metres to describe the diameter of a crater on Mars.

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.

To reach the speed of light by accelerating at 1g won't take a year, it will take nearly a year. Not a coincidence, and not even related.

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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