# In perfect vacuum (no external force), if I apply a small amount of force to a body, will it continue to accelerate forever?

According to the equation $F = ma$, the greater the force applied to a body, the greater its acceleration. But in a place where there is no gravity or any other kind of external force, if I apply a force to a body, will it continue to move forever with a uniform velocity, or will it continue to accelerate forever?

• An object only accelerates for as long as a force is applied to it. – HDE 226868 Jan 6 '16 at 4:00
• @HDE226868 Oh. So in a place where there is no external force or gravity, if I apply a certain force to a body for 2 seconds, it will only accelerate for two seconds and then move constantly with the increased velocity. Right? – arandomguy Jan 6 '16 at 4:03
• @Raj Yes, correct. After the two seconds, it will move at a constant speed. – HDE 226868 Jan 6 '16 at 4:04
• BTW, this is one of the arguments against the concept of "reactionless drive" engines for spacecraft where constant input of energy creates constant acceleration resulting in kinetic energy increasing as velocity squared – user56903 Jan 16 '16 at 10:20

The object accelerates while the force is applied to it. When the force stops acting on the object the object maintains it's current momentum (therefore it's velocity) until another force is applied to the object.

Because of e=mc^2 however, the amount of force required to for a given amount of acceleration will become greater and greater until you reacha point where you need infinite force to acclerate an object past a certain point (the speed of light).

In classical mechanics, yes. Constant force leads to constant acceleration thus the particle will accelerate forever reaching infinite velocity. If the forces stops acting, then the body continues with constant speed.

However this would lead to a speed superior to that of light ($c$) which is prohibited by relativistic mechanics. So Newton's law must be modified.

Define the vector momentum as $\textbf{p}=\gamma m_0 \textbf{v}$ where $m_0$ is the rest mass of the particle, $\textbf{v}$ is the velocity and $\gamma={1\over \sqrt{1+{\textbf{v}\cdot\textbf{v}\over c^2}}}$.

Then, as mentioned in another answers, we write the Newton's law as:

$$\textbf{F}={d\textbf{p}\over dt}$$ (which is actually the original form of Newton's law, but we usually deal with constant mass so that $F=ma$ is the same thing..).

Now assume the system is one dimensional (so no vectors!): Since velocity is also present in the $\gamma$ factor: $$F={dp\over dt}=\gamma m_0 {dv\over dt}+m_0v\gamma^3{v\over c^2}{dv\over dt}$$ which we rewrite, using $a={dv\over dt}$:

$$F=a m_0 \gamma^3({v^2\over c^2}+{1\over \gamma^2})=a m_0 \gamma^3({v^2\over c^2}+1-{v^2\over c^2}) = a m_0 \gamma^3$$

meaning $$a={dv\over dt}={F\over m_0\gamma^3}={F\over m_0}\left(1-{v^2\over c^2}\right)^{3\over 2}$$

This is our new equation. As you can see, at the beginning ($v\ll c$) it is the same as in classical mechanics. Then, as velocity increases, acceleration decreases, so that it can not reach the speed of light if not asymptotically.

One way to imagine this (though not completely right) is to rewrite this equation as

$$F=ma$$ as in classical mechanics but with a velocity-dependant mass $m=m_0\gamma^3$ (longitudinal mass. There is an equivalent transverse mass used in highr dimensions when F is not parallel to v). Thus as speed increases, the mass becomes infinite and the acceleration goes to zero.

In conclusion, the solution to our equation is $$v(t)={F\over m_0}{t\over\sqrt{1+{F^2\over m_0^2}{t^2\over c^2}}}$$ that, as you can see also from the plot (in which ${F\over m_0}=1$ and $c=1$) the speed $v(t)$ reaches $c$ for $t\rightarrow\infty$.

So speed does keep increasing but never reaches $c$! (It still is true that if the force stops acting the body stops accelerating).

Graph: x is time, y is speed.

Well the body will accelerate as long as you are applying a force on it. So if u apply a force for forever it will accelerate forever but there is an interesting thing that happens.

# Force applied for ever

You would notice that if u apply a force for an infinite amount of time then by $$F=ma$$ there should be acceleration for an infinite amount of time leading to a speed greater than light. Then is the time when you start to think that there is something wrong with this and indeed there is. Turns out that $$F=ma$$ isn't very accurate in this case but does this violate newton's second law? No, because newton's second law was never $$F=ma$$. This is a form it gets reduced to if the mass remains constant but we know that mass does increase with increase in velocity and considerably when speed reaches the speed of light.(The second law FΔt = p is still valid). So now we can see that as you reach the speed of light u need a larger force to have the same acceleration and at speeds really close to speed of light u will need an infinite amount of force to produce any acceleration in the body.