Will something accelerate forever if a constant force is applied to it on a frictionless surface? If so how?

I don't understand the fact that let's say hypothetically I was pushing a block on a frictionless surface, and I kept applying the same force on the object how it would keep accelerating! Wouldn't there have to be a certain velocity where it would stop getting faster? If not then what is actually acting on the object to cause it to accelerate infinitely?

• With questions like these, I find it helpful to ask "Why must there be a certain velocity where it would stop getting faster?" Often the issue is that constant forces and friction-less surfaces do not exist in real life, so your intuition can lead you astray. In real life, for instance, there are drag forces which increase in force as you move faster. Eventually these equal out the force pushing the block, and it stops accelerating. However, in the frictionless world you describe, these effects do not occur. – Cort Ammon Oct 10 '16 at 17:48
• Your second question is nonsensical. Your question presupposes an infinite frictionless surface and the ability to apply a force to an object no matter how fast it is moving throughout the infinite length of that surface, and you then ask "so what force is causing the object to accelerate?" You tell us! You're the one presupposing that such a force exists by supposition of the question. – Eric Lippert Oct 10 '16 at 22:20
• It sounds like you may be confusing accelerating indefinitely and increasing velocity without bounds. A massive object can always accelerate (even in the presence of drag, since a sufficient force could overcome that), but a massive object can never exceed a velocity of c (not just a good idea; it's the law). The answers explain why. – Aaron Novstrup Oct 11 '16 at 0:06
• "what is actually acting on the object to cause it to accelerate infinitely?" - you are! You said so in the first sentence! – user253751 Oct 11 '16 at 0:19
• I think a simpler way of explaining it would be that a constant force applied to the object would indeed continue to accelerate it. But the faster it goes, the less acceleration would be extracted from the same amount of force (because, relativity or something), so it would level off as you slowly (but never actually) reach $c$. You'd have to increase the force itself exponentially in order to keep your acceleration constant, and then you'd eventually need more force than is physically possible once you hit $c$, hence why it's the universal speed limit. – thanby Oct 11 '16 at 18:36

Yes, an object of mass $m$ subjected to a constant net force $F$ would continue to accelerate acc. Newton's second law:

$$F=ma$$

Where $a$ is the acceleration, aka the rate of change of velocity $v$ in time.

But when the velocity $v$ starts approaching the speed of light $c$, Newtonian physics no longer applies and we need to apply Einstein's theory of relativity. This prevents the object from exceeding the universal speed limit, $c$.

• But if you keep applying the force, then object will keep accelerating. But the from an "stationary" observer the object's speed will seem to asymptotically approach the speed of light, due time dilation. But the object keeps accelerating, which also "causes" it to approaches the speed of light, but the "stationary" space around it will experience length contraction. So the object would be able to travel a light year (measured from a stationary observer) in less than a year (measured from the object). At least this is my understanding of relativity. – fibonatic Oct 10 '16 at 20:17
• My psychic powers tell me OP is not asking a question about Newtonian vs. modern physics, but rather a question about Aristotelian vs. Newtonian physics. If my guess is correct, then none of the current answers would appear to be relevant. – Kevin Oct 11 '16 at 3:27
• If in the thought experiment you do have a frictionless surface and you can keep applying acceleration even as it approaches c. What exactly keeps it from breaking c? If there is no friction doesn't that solve the infinite mass problem? (my understanding was that anything with mass approached infinite mass as it approached c, which is why it couldn't go faster. I'm programmer, not physicist, sorry.) – Ryan Oct 11 '16 at 20:19
• @Ryan: have a look at this: en.wikipedia.org/wiki/Mass_in_special_relativity – Gert Oct 11 '16 at 20:54
• @Ryan An object of near-infinite mass cannot be significantly affected by a finite force regardless of whether there's friction or not. – Ajedi32 Oct 11 '16 at 21:20

When special relativity is taken into account, for an object accelerated at constant rate $a$ with respect to its instantaneous rest-frame, then as a function of proper time $T$, the ratio of the rocket's velocity $v$ (with respect to its initial rest frame) to the speed of light $c$ is given by

$$\frac{v}{c} = \tanh\left[T \left(\frac{a}{c}\right)\right]$$

So the graph of velocity versus proper time looks like this:

The velocity initially increases linearly with T, as you'd expect in the non-relativistic case, but eventually the velocity asymptotes at $c$.

• Exactly where does the linearity stop? – bpedit Oct 10 '16 at 18:43
• Exactly, as the equation in the answer shows, it is never linear. Just looks approximately linear for low v/c – Bob Bee Oct 10 '16 at 19:03
• @BobBee I thought it is linear when $v = 0$. – dtldarek Oct 11 '16 at 13:07
• @dtldarek The linearity near $v=0$ is only "good approximation". In other words, it seems linear, because the nonlinearity is overwhelmed by the error of the measurement. – Crowley Oct 11 '16 at 15:17
• It shall be noted, that the acceleration never stops, the speed of light is a limit and the actual speed of the object never stops approaching it but never reaches it either. – Crowley Oct 11 '16 at 15:20

One way to think of this scenario is to imagine the pressure on your finger (I'll assume you're pushing with your finger). If you are pushing with a constant force the pressure on your finger will remain constant. In practice as the object accelerates you will have to move your finger faster and faster in order to 'keep up' and maintain this pressure. Remember that in this thought experiment that the pushing entity must accelerate just as fast as the object in order to maintain the force.

• How fast is the earth moving then when it accelerates me to the ground after I jump off a bridge? – whatsisname Oct 11 '16 at 21:31
• @whatsisname In your frame of reference, the Earth rushes towards you with an acceleration of ~9.80665m/s². – mg30rg Oct 12 '16 at 12:34
• @whatsisname It's not moving; that's why, if you passed through the Earth instead of hitting the ground, the force on you would drop to zero as you reached the center of the Earth, and would then grow in the opposite direction until you reached the other side. The gravitational acceleration imposed by the Earth is not constant everywhere because the Earth is a finite size; to accelerate an object indefinitely the Earth would have to be of infinite size, or move along with the object. – 2012rcampion Oct 12 '16 at 18:33
• @mg30rg: and in the finger scenario, in that frame of reference the finger has a motion of 0 m/s. So which is it? Or perhaps that aspect of the scenario is not actually important........ – whatsisname Oct 12 '16 at 20:36
• No specific frame of reference is required for the 'finger scenario'. Note that the finger is accelerating. It will not have a constant velocity in any frame of reference. – mattfitzgerald Oct 12 '16 at 23:49

You seem to be confused because this contradicts everything you know about physical objects. That's a reasonable response! A box continually speeding up while being pushed sounds preposterous.

However, if you could continually push the box, and if there were no friction slowing it down, it would continually accelerate. (Eventually you would observe relativistic effects as its speed became comparable to the speed of light.)

Two parts of this hypothetical situation are extremely strange: the continual force, and the frictionless surface. Both of these are far outside anyone's experience. Usually a rocket will run out of fuel and stop accelerating, and usually friction will become stronger and stronger until the object slows.

A spaceship that had a ramjet (a hypothetical engine that collects the sparse hydrogen of space and uses it as limitless fuel), flying in deep space with essentially zero friction, would act that way in reality. It would continually accelerate without limit. (as it approached lightspeed, it would increase in mass instead of velocity, but I don't think you're asking about relativistic effects.)

• It seems that a Bussard Ramjet actually does have a top speed limit - the collection of fuel and propellant will eventually cause more drag than you can overcome. – Luaan Oct 11 '16 at 7:45
• The infinite extent of the surface is also far outside anyone's experience. – David Richerby Oct 11 '16 at 11:07
• Isn't it so that a rocket can only apply force when the propelent is exiting the rocket faster than the object is moving? – Roy T. Oct 12 '16 at 8:39
• @RoyT. No. Ejecting any amount of mass from the back of the rocket at any positive speed (relative to the rocket) will accelerate the rocket, by conservation of momentum. – David Richerby Oct 12 '16 at 9:49
• @DavidRicherby ah of course, forgot about that :) – Roy T. Oct 12 '16 at 9:52

What you're asking about is described as a Rindler coordinate system. So, the answer is: yes, the acceleration is always positive. The energy is always increasing. And different inertial observers will have different perspectives on the world-line traced out by a Rindler observer.

What you're describing is basically a newtonian orbit. Look at the Moon, for example - it is being continually accelerated by an almost constant magnitude force. That's how it stays in orbit in the first place :) Acceleration doesn't necessarily mean that the magnitude of your velocity is increasing, just like applying force doesn't necessarily mean acceleration.

General relativity has a different look on the problem - in GR, there is no acceleration on an orbital path. Relativity gives you a few more kinks as well - for example, the maximum relative velocity you can have is a tiny bit shy of the speed of light. Despite this, if you had a space ship with a magical propulsion system, you could continue accelerating indefinitely - you on board would feel constant acceleration, while somebody observing your flight from Earth would just see your velocity approach the speed of light slower and slower.

• I suspect the asker is imagining that the force is parallel to the motion, not perpendicular to it. Not least because a planet in its orbit is operating within a constrained range of speeds, despite the continual force operating on it. – David Richerby Oct 11 '16 at 11:11
• @DavidRicherby Yeah, that was just an example that continuous acceleration doesn't necessarily mean an increase in speed. The continous increase in speed part is addressed in the second part of my answer. – Luaan Oct 11 '16 at 12:10

Consider that instead of pushing the block in the direction it was moving you could push it perpendicular to its motion. This would cause it to change direction (which is still acceleration) without speeding up. The result would be a circular motion which would accelerate infinitely.

Let's analyse Newton's law as it was proposed: $$F=\frac{dp}{dt}=\dot p=\dot mv+m\dot v=\dot mv+a$$

• In non-relativistic system $\dot m=0$ and all the force is "consumed" in acceleration.
• In relativistic system the closer the $\frac vc$ is to $1$ the more significant the $\dot mv$ term is relative to $m\dot v$. $F>0\Rightarrow \dot mv>0\ \wedge\ m\dot v>0$, where $F$ is the net force (friction is compensated).
• For Speeds close to the speed of light $$\lim_{v\rightarrow c}\dot mv=F \text{ and }\lim_{v\rightarrow c}m\dot v=0$$

The body will accelerate untill... No, it never stops accelerating. And it never reaches speed of light.

Theoretically, it would get infinitely close to C, where it would require more and more energy to accelerate the same amount, and if your friction less surface is long enough and you can somehow keep applying this force to the block.