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say someone punched the ground (Earth) with infinite strength, how much force would they require to destroy/ vaporise the earth? Or an easier question how much force would someone need to induce $3.2 \times 10^{29}$((approximate recall) Joules, which is enough to unbind the Earth), to the Earth?

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    $\begingroup$ This may be a simple matter of scaling up the calculation at what-if.xkcd.com/1. A baseball is similar to a fist. Apparently at 90% of the speed of light you get destruction similar to a nuke. Thanks to relativity you can pack as much energy into the punch as you like and you don't even have to exceed the speed of light. $\endgroup$ Commented Jun 3, 2022 at 16:46
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    $\begingroup$ I thought Teller and Bethe proved in the 1940's you can't ignite the atmosphere even if you set off a nuke, ie, not a firecracker's worth of fusion. $\endgroup$ Commented Jun 4, 2022 at 0:37
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    $\begingroup$ The superpuncher would need more than infinite strength, he would need some supermeans of keeping himself in one place and supertoughness to handle delivering the punch. Also he would need to reach bedrock first. $\endgroup$ Commented Jun 4, 2022 at 21:49
  • $\begingroup$ If you someone really punched the ground (Earth) with infinite strength, that would clearly be greatly more force than was needed… making the Question, as worded, irrelevant or pointless… take your choice. If you mean simply, how much force would be required to destroy the Earth, why not ask that? If you mean "how much force would be needed to induce 3.2×1029 (approximate recall) Joules" why not Ask that? Unless you don't mean "how much force would be needed to induce 3.2×1029 (approximate recall) Joules" why not take the next steps in maths? $\endgroup$ Commented Jun 5, 2022 at 19:15

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It cannot be done through a punch. There's several issues, but the easiest to work with is to look at the displacement of your fist. From your invocation of the binding energy of the Earth, it doesn't look like you're interested in some garden variety "destroy the planet" sort of punch. You're looking for something which pulverizes the earth and sends its ashes off to the corners of the cosmos. That's some next level damage there! Okay, let's calculate!

I'm going to assume that this is less of a "punch" and more of a "shove." Why? Because Work = Force * Distance. Work is what you need to put into the earth to get the energy needed to overcome the binding energy. Speaking of which, the energy is slightly higher than you recall. The gravitational binding energy of the Earth is $2\cdot10^{32} \text J$. Minor detail. It will turn out 3 orders of magnitude isn't much in this problem! Now an average arm span is 1.4m, we'll round up to 1.5. Might as well give them every advantage we can get! Dividing $\frac{2\cdot 10^{32}\text J}{1.5m}$ yields $1.3\cdot 10^{32} \text N$

Now this is where it's going to get murky. Understand just how absurdly large that force is. That's the force of 3,800,000,000,000,000,000,000,000 Saturn V rockets all lifting off simultaneously. I wanted to find a comparison that involved fewer zeros, but my favorite "Orders of Magnitude" series of pages in Wikipedia did not have much in that range to work with. It's quite literally 10,000,000,000 times more powerful than the force the sun applies to the Earth. This is a bonkers level amount of force.

Of course, this assumes you can apply it. A few years back, I fielded a question on Physics.SE on this topic. Just because you are capable of generating such forces in theory doesn't mean they actually materialize. Once you start getting into high enough forces to break the object you are punching (read: pulverizing granite), the electrostatic forces holding it together fail, and your force starts accelerating the mass in front of it rather than distributing that force across the rest of the wall. What you would find very quickly is that your fist simply pulverizes the nearby slug of rock and you fail to actually impart this energy.

Which brings us to the world of supersonic propagation. You need to be moving fast enough to impart that kind of energy to only the matter in front of you. There will be some spread, but as we will see, at the speeds we are talking about here, it's really just a cylinder of rock we need to consider. We are considering the rock that is physically occupying the path your fist takes. That would be 1.5m long, from the armspan before, times $0.054\text m^2$. Why that number? That's the surface area of an outspread hand. A fist is slightly smaller, at $0.045\text m^2$, which makes the story worse, so we're going to try to slap the Earth out of existence rather than punch it. Multiply these through, we see that the cylinder of rock your hand can physically reach is $0.081 \text m^3$. The density of diabase is $2970\frac{\text{kg}}{\text m ^3}$. Diabase is a very hard rock, harder than granite, and slightly more dense, so it seems like the best surface to apply your punch to. This means your fist has direct access to $240 \text{kg}$ of rock. Using $E=\frac 1 2 mv^2$, we find the velocity that rock needs to reach to have the necessary energy is $1.3\cdot 10^{15} \frac{\text m}{\text s}$.

Now we have a real problem. The speed of light is $10^8 \frac{\text m}{\text s}$. By these simple Newtonian calculations, we would need to strike the earth at a velocity equal to 10,000,000 times the speed of light. Bollocks. This obviously can't work, which means now we need to toss away the Newtonian calculations and start using relativistic ones. Incidentally, this means your fist is now less of a first, and more of a concentrated burst of high energy particles.

Relativistic mass and "rest" mass are related by the equation $m_{rel}=\frac{m_{rest}}{\sqrt{1-\frac{v^2}{c^2}}}$. Remember, we need that slug of rock to have all of the kinetic energy needed to destroy the Earth, so our $m_{rest}$ is the mass of the rock. We can also use the famous equation $E = m_{rel}c^2$ here. We need the rock to have the requisite energy so its relativistic mass must be $2.2\cdot 10^{15} \text{kg}$. Thus $\frac{m_{rel}}{m_{rest}} = 9.25\cdot 10^{12}$. The relativistic effects need to provide a mass multiplier of just shy of ten trillion to 1.

How fast does this need to be? My calculator actually runs out of precision, but it is at least 99.999999999% of the speed of light. That's at least 10 nines. For perspective, the LHC throws electrons around at 99.9999991% of the speed of light - 8 nines.

So, at this point, we really do need to start asking tough questions about whether this qualifies as a punch. Its more of a particle burst. And the question I held off till the end: how did you accelerate your hand like this? To impart this much energy into the Earth, the equal and opposite reaction is going to send you careening off at nearly the speed of light. So, in the end, we destroy the planet, and the super-villain!

Now, just for fun, a supernova outputs about $10^{44} \text J$, in all directions ($4\pi$ steradians). This means that if the Earth subtends a solid angle of $10^{-12}$ steradians, that supernova would strike the Earth with the energy you so crave. This would put the supernova at a distance of about $10^{12}\text m$ away. In other words, for one of the most furious events in all of astrophysics to impart that kind of energy into the Earth, it would need to be somewhere around the distance to Saturn. A supernova... inside our own solar system...

"It must be Thursday. I never could get the hang of Thursdays." -Arthur Dent

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    $\begingroup$ I am deeply reassured that we are effectively safe from a Superman-like-being throwing a punch (slap) that destroys the Earth - the artists who do comics would never be able to accurately draw a hand behaving as a concentrated burst of high energy particles. +1 for an answer worthy of XKCD What If. $\endgroup$ Commented Jun 3, 2022 at 3:47
  • $\begingroup$ +1 for the Arthur Dent reference, +10^6 for the actual answer (if I could) 😊 $\endgroup$ Commented Jun 3, 2022 at 10:31
  • $\begingroup$ One Punch Man can still do that with a punch. *ducks away* $\endgroup$
    – orithena
    Commented Jun 3, 2022 at 10:38
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    $\begingroup$ @ilkkachu perhaps their measure is not the face of the hand, but actually the area of all the skin wrapped around it, front to back. $\endgroup$
    – Drake P
    Commented Jun 3, 2022 at 12:56
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    $\begingroup$ I like the calculation but I don't get the pessimistic attitude. "10 billion times more powerful than the sun"? A being of infinite strength will not even break a sweat. "That's at least 10 nines." They have to be careful not to reach that speed with every step they take. $\endgroup$ Commented Jun 3, 2022 at 16:43
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At this point, you would need to be more concerned with recoil of that punch. However force it may take to shatter the earth, the some ”one” would probably be yeeted back at light speed. That would honestly be really funny.

Not to mention, even if this person punches earth with supposed infinite force, that would probably won’t do shit. Major limitation is the form factor of puncher and that fact that material making up earth ain’t that tough.

The length of a human arm is somewhere in between 60-90 cm. When the punch that powerful connects: rocks and hard surfaces will break and soil will compress, and punch will displacement them further deep down or sideways.

So all the damage you can do is with displacement of earth caused by your arm, roughly equal to volume of the arm (+/- some cc), the associated shockwave.

Depending on actual speed of punch, all can do is make a crater of 1-1000m in radius and much less in depth and a short earthquake.

In human form, I believe it’s impossible unless you hit at light speed and cause destruction due to light speed/relativistic shenanigans.

You would need to make the fist itself big in size such that damage is more spread across the planet. Your punch may be only need to severely damage or essentially bore up to 100-600 km (probably less) deep into earth and basically expose the mantle, and internal pressure of earth cause it fall apart on its own by.

And shockwaves from punch that deep actually may be strong enough break the planet or at the very least cause some serious instability within the earth.

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  • $\begingroup$ Thanks, that was my thought, too. The absolute worst outcome would be equivalent to a baseball sized chunk of antimatter hitting the planet, and that's been calculated in a few places. $\endgroup$ Commented Jun 2, 2022 at 18:58

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