# What would prevent object from experiencing an infinitely powerful jerk in an instant?

Unlike speed which is capped for anything with rest mass at speed of light in a vacuum, what would prevent an object to undergo infinite acceleration in an instant? I assume in theory if we can apply infinite amount of force on the object at a particular moment, it must undergo an infinite acceleration right? What laws of physics work to prevent infinitely powerful jerk from happening?

• Objects undergo infinite acceleration in elastic collisions and the physics is fine with it. – Sam Jul 2 '20 at 4:49
• @Sam No, they don't. It's just another assumption in the list of assumptions used to simplify the physics involved in the collision. – user258881 Jul 2 '20 at 5:09
• @FakeMod So what happens in an ideal elastic collision then? – Sam Jul 2 '20 at 5:53
• @FakeMod What about a massless particle, such as a photon being reflected? – Anders Gustafson Jul 2 '20 at 6:28
• @Sam There is a short amount of time ($\approx 1\:\rm ms$) when both the balls are in contact with each other, where one is decelerating and other is accelerating, both at a finite acceleration. – user258881 Jul 2 '20 at 13:13

Note: The following answer has been given from a classical mechanics point of view. It does not deal with any quantum mechanical phenomenon arising due to the following scenarios.

## Theoretical analysis

An infinitely powerful impulse can, theoretically, exist. Because, mathematically

$$J=\Delta(mv)=\Delta p=p_{\rm final}-p_{\rm initial}=p_{\rm final} \qquad \text{(when }p_{\rm initial}\text{ is zero)}\tag{1}$$

where $$J$$ is the impulse applied. Now since the momentum considered here has to be relativistic momentum, thus the formula $$(1)$$ gets modified to

$$J=\gamma m v=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\tag{2}$$

Thus, according to formula $$(2)$$, as $$v\to c$$, the momentum (and thus the impulse needed) blows up to infinity. Thus you could easily apply an near infinite impulse and still never reach the speed of light. Now since the force applied can be written as

$$F=\frac{ \Delta p}{ \Delta t}=\frac{J}{ \Delta t}\tag{3}$$

So if you manage to accelerate an object to a very high (near light) speed in a finite interval of time, you would be applying as large force as possible. Thus, the closer the object's speed reaches to the speed of light, the larger the impulse (and the corresponding force) you'd need to apply.

However, since an object can never attain a speed equal to $$c$$, thus the impulse can also never reach infinity. Thus, mathematically $$J\in[0,\infty)$$.

## Practical analysis

### Time

Note: The following analysis might sound a bit philosophical/metaphysical, however I see no reason for it to be wrong

Now if you're wondering that if we could make the $$\Delta t$$ extremely small, to make the force infinite, then... No. Not really. Time, as we perceive, is continuous. You cannot choose any two different unique time instants separated by a zero time difference. It's the same as saying that there are no two distinct real numbers whose difference is zero.

You might argue that why don't we choose the same time instant instead of two different time instant. This is not possible under the domain of classical mechanics, since one of the fundamental assumptions of classical mechanics is that the state of any system is unique at a certain instant of time. Thus you cannot have finite $$\Delta p$$ for two same time instants. For more insight, see this PhysicsSE question.

### Practicality

Practically speaking, I can quite confidently say that there are no man-made machines which can cause any sort of infinite acceleration, and neither are we gonna make any of them. Despite of the unphysical-ness of such infinite acceleration, I reckon we are ever going to have enough energy to accelerate something heavy even to anything near infinity.

Conservation of energy certainly does not allow $$\mathrm dp/\mathrm dt$$ to be infinite. $$p=mv$$ and this means kinetic energy will be infinite. If you set $$\Delta t$$ to zero while keeping $$\Delta p$$ finite it is a condition not realizable in bulk matter, one has to go at the microscopic level where quantum mechanical uncertainties enter ,

\begin{align} \Delta x \Delta p&>\frac \hbar 2\\ \Delta E \Delta t &>\frac\hbar 2 \end{align}

• That doesn't make sense, $p$ can be discontinuous (infinite time derivative) while being perfectly finite so that the kinetic energy never becomes infinite. – KF Gauss Jul 2 '20 at 4:09
• @KFGauss I consider this in my second sentence (forbulk matter how will you measure delta(t) equal to zero?) – anna v Jul 2 '20 at 4:11
• Any object that has mass cannot experience infinite acceleration because that would require an infinite force, and there are no infinite forces in nature. – niels nielsen Jul 2 '20 at 6:14
• @nielsnielsen I agree, I am saying the same in different words – anna v Jul 2 '20 at 6:46

Well, simply put: $$v = \int a dt$$ so if you increase the acceleration you increase the velocity. But, $$a = \frac{F}{m}$$ and very loosely speaking, as the velocity increases the mass (more precisely, the inertia does) increases, approaching infinity. So even an infinite force (which cannot exist) cannot ignite an infinite acceleration. So, physics cannot allow infinite acceleration, because the velocity would increase too, but then it cannot increase higher than a certain level.

• "...physics cannot allow infinite acceleration, because the velocity would increase too, but then it cannot increase higher than a certain level." This is wrong. Force is more precisely defined as $F=\mathrm dp/\mathrm dt$. According to special relativity, the relativistic momentum is given by $$p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}$$ As you can see, $p\to \infty$ when $v\to c$, so you can apply an infinite force (to produce an infinite momentum change), and still never attain infinite velocity (or even $c$) – user258881 Jul 2 '20 at 13:19
• But what I said is that the infinite force could not cause infinite acceleration, because of the increase in inertia. – PNS Jul 2 '20 at 13:30
• That's also incorrect, IMO. Because if an infinite force existed, it would definitely cause an infinite acceleration, however it cannot sustain that acceleration for any finite interval of time :-) – user258881 Jul 2 '20 at 13:53