The maximum speed the planet could accelerate you to, assuming you're starting at rest, is the escape velocity of that planet.

To truly reach that velocity, you'd have to start at a distance of "infinity" away from the planet. Obviously, you can't start at infinity, but even starting out relatively close- like the distance to the Moon, gets you within 99% of Earth's escape velocity.

What is kind of cool is, the way you calculate escape velocity is basically the exact same calculation to calculate the maximum speed a body could get you to. It all comes down to conservation of energy. The argument goes: Energy is conserved. If you start at an infinite distance away from the Earth, at rest, your total potential energy is zero, and your kinetic energy is zero. So, as you fall towards Earth, your total energy must stay zero. Since kinetic energy is positive, and potential energy is negative- the closer you get to Earth, the more negative your potential energy must be, and thus your kinetic energy has to grow to keep it at zero. So, at any distance, r, from the Earth, if you started at infinity and moved towards the surface, you could calculate your speed as follows:

1/2mv^2 - GMm/r = 0

cancel m and move over the potential energy term

1/2v^2 = GM/r

Solve for v

v = sqrt(2*G*M/r)

If I may ask, the rule/law that your energy must be zero is the Law of Conservation of Energy right?

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Not quite. There is no law that says the energy must be zero. That was just a starting assumption in order to arrive at some kind of answer.

What Conservation of Energy says is that the total energy (potential + kinetic) does not change as the body moves toward Earth. If it starts at zero energy, then the energy remains zero. But the energy could have started with some other value too.

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There is no way I could trade math equations with you; but if I’m falling directly at the planets core; how is escape velocity relevant? I should continue to accelerate- ever faster as the separation decreases, until the planet’s surface stops me.

That's true. You will continue to accelerate until hitting the surface. And the speed you'll be accelerated to is the escape velocity. And it's because your starting and end states are the same, just in reverse.

When calculating escape velocity you're saying "how fast do I need to launch myself from the surface of the planet, so that I don't come to rest until infinity?" But for this question you're saying "if I'm at rest at infinity, what speed will I have when I hit the surface of the planet?" Since the start and end points are the same, the speeds are also the same.

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No, that's not true... If you couple the object to a mechanism that extracts energy while keeping the speed under terminal velocity, then you may recover more energy from the potential energy than this limit you're imposing. For example airplanes (coasting) convert potential energy into kinetic energy by purposefully staying below terminal velocity

Here I'm only counting "useful" energy. But you're ignoring the potential energy converted to air heating and turbulence along the way when the object is at terminal velocity

You're 100% right. I forgot that it wasn't a closed system. I'll sit in the corner with my dunce cap on for a while.

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So that's basically more than the height of the mount everest per second.

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More accurately, it’s just that you’ve defined you’re potential to be zero at infinity. You could’ve defined it to be anything (finite) at infinity so long as you remember to add that to the energy you have at radius r too

There’s important distinction between assuming and defining!

If the force that accelerated you were constant, you could indeed reach a speed arbitrarily close to the speed of light just by starting arbitrarily far away – and escape velocity wouldn't be a thing, either.

But gravity is inversely proportional to distance squared, so if you're very far away (in a toy universe where there are only you plus the object you're falling towards), your initial acceleration will also be very slow and almost all of the speed gain will happen when you're already very close to the object.

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In case people want to put some numbers to that and are too lazy to look them up, that's 11.2 km/s or 40,000 km/h. For americans, that's 25,000 mph.

If you're starting from rest and falling towards a planet, then you will accelerate faster until you hit. Remember that gravity gets lower the further away you are and eventually you will be far enough away that the planet's pull on you isn't relevant anymore - escape velocity is how fast you would accelerate to if you started at the point where you are only just falling towards the planet (simplifying a lot but close enough)

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Wait, why is it that it can never be greater than escape velocity? If theyre falling towards the earth then surely thats not an issue as they would hit the planet not escape

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