Figuring out the math for the Sphere of Meteoric Conversion

Question

I'm presently in the process of developing my plans for a flintlock fantasy series. It's high magic, but one form of magic in the setting, Arcane Magic, is heavily based in science. The energy that powers it, Aethyr, can bend the laws of physics but not outright break them. (You can't turn someone into a frog with Arcane Magic, in other words.) My protagonist, Perdita, is an Arcane Engineer. She's someone who makes magical devices of an Arcane nature. Among her creations is a belt that can perform a Spell called the Sphere of Meteoric Conversion. Here's the basic idea of how it works:

When Perdita falls beyond a certain distance, the belt activates, using Aethyr to create a magical bubble around her. When she hits the ground (or any solid object larger than the average bird,) the Sphere of Meteoric Conversion converts all her kinetic energy and the kinetic energy of the surface pushing back against her (that is, the equal and opposite reaction) into another kind of energy or simply redirect the kinetic energy away from her.

By this I mean that the bubble absorbs all the energy of the impact and transfers it away from Perdita so she takes no damage, either by directing it at something other than her or by converting it into a different kind of energy which is then directed away from her.

So, the bubble may just cause all the absorbed kinetic energy to go outward from it in all directions, like a bomb going off. Or it may convert all the kinetic energy into thermal or radiant energy and release it outward in all directions. I suppose converting it into sound is also something it can do.

The Sphere of Meteoric Conversion can also direct the absorbed and converted kinetic energy in a ring or a beam. For example, when she hits, the bubble discharges a disk of light, heat, or sound along its proverbial equator. Or it may discharge a heat beam or laser beam directly up into the air. The point is that it takes the energy of the impact and turns it into something else to be used in a different way while Perdita remains safe inside the bubble.

And, before you ask, yes, the amount of Aethyr needed to create the Sphere of Meteoric Conversion is considerable, especially if she's moving at terminal velocity. The belt is going to need some time to "cool down" (not literally) before it can generate the Spell again.

Also, the Spell doesn't have to convert all the kinetic energy into just one other type of energy. It can convert some of it into radiant energy, some of it into thermal energy, some of it into sound, and just redirect any left over kinetic energy away from Perdita.

What I need help with is determining the math for the Sphere of Meteoric Conversion. I want to make sure the math holds up in the books. Also, is there anything I'm overlooking or missing with this idea? I know it's magic, but, like I said, Arcane Magic is intended to be as science based as possible.

It's going to be interesting seeing the feedback I get on this. I hope that it presents some of you with a fun puzzle, at the very least.

Edit and Update

Okay, it seems I need to clarify another point here: The purpose of the Sphere of Meteoric Conversion is to weaponize the kinetic energy of the impact in addition to preventing Perdita from taking any damage. Alternatively, it will convert the kinetic energy into something harmless so she doesn't hurt herself, her allies on the ground, or do any significant property damage. I think it will be easier to explain this with a few examples. Some things that will be applicable in all the examples are the following:

Perdita is a Half-Orc, not a Human. She's 6'7" tall and weighs about 220 lbs. The gear she usually has on her probably comes to a total of 30 lbs., minimum. (That's her leather garments, melee weapons, flintlock guns, ammunition, etc.) So, total weight is around 250 lbs.

Example 1: Perdita is up on the tower of a castle. In the courtyard below is a horde of Ghouls (basically zombies) trying to break into the front entrance of the main keep. Perdita has the Mage with her use a Spell to launch her high into the air so she then comes down in the center of the mob of Ghouls. The Sphere of Meteoric Conversion changes the kinetic energy of the impact into thermal radiation and releases it as a disk from the "equator" of the bubble. Ghouls are weak against extreme heat. Or she may have the thermal radiation released as a beam that targets one really big Ghoul that's the size of the Incredible Hulk. The point is that she uses the kinetic energy of her impact to burn some Undead.

Example 2: Perdita is on a floating island over the ocean. She gets blown off it, either by a powerful burst of wind or a Mage using a Spell. As she falls towards the water below, she see a Sea Monster rising up, its jaws open to catch her. She activates the Sphere of Meteoric Conversion and adjust it to its "Zappy" setting. When she lands in the Sea Monster's mouth, the kinetic energy is converted into a pulse of electricity.

Example 3: Perdita is on an airship and, for one reason or another, gets thrown off of it. As she falls to the ground, she sees that she is headed toward a park where an aristocratic girl is having her Quinceañera. And the birthday girl has just received a puppy as her gift from her parents. Not wanting to become a party crashing in the most literal sense of the term, Perdita activates the Sphere of Meteoric Conversion and adjusts it to its "Harmless" setting. On impact, all the kinetic energy is converted into a beam of light (perhaps visible light, perhaps radio waves or something that won't hurt anyone.) The only damage done to the festivities is in the form of extreme awkwardness.

These examples are hopefully sufficient to explain the the why of the Sphere of Meteoric Conversion. Perdita isn't just trying to protect herself from injury and death. She also wants to harness the kinetic energy of the impact in some way or just direct away so nobody is hurt and no property damage is caused. If she just wants to slow her fall so she lands safely, she'll use a different kind of Spell, but there are situations where a slow descent would be a problem, like if that would make her an easy target to shoot from the ground or if that would give someone she's chasing time to escape and hide.

Hopefully this clarifies what I'm trying to figure out with this post. Also, having some examples of how to use the equations provided would be helpful, since I'm not a physicist and don't know what all the symbols in some of the equations mean without being told first. Once I do know, then I can use the equations myself, however, so I'm not asking other people to do all the calculations for me. I hope this clears things up.

Answer 1

When Perdita falls beyond a certain distance, the belt activates,
using Aethyr to create a magical bubble around her. When she hits
the ground (or any solid object larger than the average bird,) the
Sphere of Meteoric Conversion converts all her kinetic energy and
the kinetic energy of the surface pushing back against her (that
is, the equal and opposite reaction) into another kind of energy
or simply redirect the kinetic energy away from her.

There are two problems with this:

1. The ground doesn't have kinetic energy separate from or opposite to Perdita. And exactly how much kinetic energy there is depends on what frame you are doing the calculation in. The relevant frame for energy dissipation in a collision is the center-of-mass, zero-momentum frame of the colliding objects; when one of those object is the ground, the center-of-mass frame is for all practical purposes indistinguishable from the ground frame, and all of the kinetic energy is Perdita's.
2. The ground does exert an equal-and-opposite force on Perdita, thus transferring momentum to her and accelerating her to match velocities with the ground. It is that process which is responsible for injury and death from falls. Energy isn't irrelevant, but it's not the major player here; dissipating energy is easy. Limiting acceleration is key to safety. Perdita could absorb all of the kinetic energy of a terminal-velocity freefall into her own body as heat and be no worse for it, but that's no good if differential acceleration turns the physical structure of her body into jello.

So, the bubble needs to do some combination of three things:

1. Slow her descent, spreading the acceleration over a longer distance to keep the g forces survivable. This is what, e.g., crumple zones in cars do--they crush so that the body of the car, containing squishy passengers, can continue moving for a few more feet after a crash, over which distance it can slow more gradually. This is also what airbags are for--they ensure that the passenger's body begins accelerating before hitting the steering wheel / dashboard / windshield, rather than doing at all at once on impact with something hard and unforgiving.

2. Magically stiffen the materials of her body so that all parts of her experience near-identical accelerations, and thus are not torn apart by differentials in momentum transfer. This kind of what g-suits do--by applying compressive forces to certain parts of the body, they ensure the g-load is distributed more evenly over the whole body, thus helping to prevent blackouts. This is where intentional energy dissipation becomes relevant; if the bubble and everything in it is super-stiff, it will just bounce. Perhaps the bubble could have multiple layers; an inner layer that briefly freezes Perdita's body, and an outer layer that slows the inner layer down over some finite distance and dumps the kinetic energy into heat. Like dropping a frozen egg inside a balloon full of jello.

3. Automatically re-orient Perdita's body into the most g-tolerant position. People can survive much higher accelerations on their backs than from any other direction, which is why infant car seats are rear-facing; not only does it help spread the load of a crash over a wider surface area compared to straining against relatively thin straps, but even controlling for equal support, the human body is simply more resistance to injury in that orientation.

The relevant math is as follows:

$KE = \frac{1}{2}mv^2$

Suppose Perdita weighs around 75kg (a reasonable number for a healthy adult human woman). Terminal velocity for people is somewhere around 55m/s. That means she has to dissipate $\frac{1}{2}(75kg)(55m/s)^2 = 113,437.5 J$ of energy, or around 27.1 food Calories. I.e., enough energy to raise 27kg of water by 1 degree celsius. Or all of Perdita's body by considerably less.

Meanwhile, people can easily be seriously injured (or, if old or just really unlucky, outright killed) by low-velocity falls just from tripping on the sidewalk--it's not how much energy is involved, it is how that energy is applied in conjunction with momentum.

$p = mv$

Perdita's momentum is $4125 \frac{m\ kg}{s}$

$a = \frac{v^2}{2d}$

Above an acceleration of 3g, Perdita will be uncomfortable. Above about 50g, she will sustain injury regardless of orientation. Above around 10g, you need to start employing those other measures like automatic reorientation and magically redistributing forces to avoid serious injury or death from falling the Wrong Way. So, if we want to simply dissipate energy into heat by slowing Perdita's descent, the bubble needs to have area of effect large enough to start slowing her descent at least $d = \frac{v^2}{2a} = \frac{(55m/s)^2}{6g} \approx 51.5m$ So, well above rooftop height if she's already at terminal velocity. At 10g with automatic reorientation, you're looking at a 15-meter bubble.

If the bubble itself supplies additional air resistance, that terminal velocity will come down a bit, which helps. But if you want Perdita to land comfortably, on her feet so she can walk away (and maybe to an awesome superhero power-pose), and without the bubble interfering with the landscape in a 100-foot wide circle all around, I think you're pretty much gonna have to look at the "magically freeze her solid" / "Star Trek inertial dampeners" option.

EDIT, to account for the question edit:

Perdita is a Half-Orc, not a Human. She's 6'7" tall [...].
So, total weight is around 250 lbs.

250lbs is about 113kg. So, applying the kinetic energy equation again, and assuming she's still falling at a typical terminal velocity for a skydiver, she's got a total kinetic energy of $\frac{1}{2}(113kg)(55m/s)^2 = 170,912.5J$ To convert joules to kilocalories (of "food calories", where 1 kilocalorie is enough energy to heat a kilogram of water by 1 degree--a decent approximation for how much you will heat a squishy meatbag), just divide by 4184. Ergo, Perdita has access to about 41 (rounding up) kilocalories, enough kinetic energy from her fall to heat 41 kilos of ghoul by 1 degree. That's not much. She will not be incinerating any undead with that.

However, once again we have to note that what matters is not really how much energy is available, but precisely how it is applied. Bullets have far less energy than that, but they're still lethal. If all that energy is released as an instantaneous flash of isotropic radiation, you won't incinerate or even set fire to any ghouls, but you might give them some nasty sunburns as most of the radiation is absorbed in the first few millimeters of skin / rotting flesh. That may or may not slow them down, depending on precisely how your flavor of undead happen to work.

And if you concentrate that energy into a coherent beam, well... you can heat 1 gram of water by 40,000 degrees (not really, 'cause there's a phase change to worry about after a mere 100 degrees max, but that hardly matters at this scale). In other words, you can flash-vaporize a tiny chunk of ghoul flesh, causing it to violently explode as if it had been hit by a bullet. Or several small chunks, from several different ghouls, if Perdita has enough control to direct multiple beams like that.

Enough energy to make the ghouls uncomfortable from extreme heat? No way. Enough energy to seriously mess up their day through other means? Absolutely.

She activates the Sphere of Meteoric Conversion and adjust it to its
"Zappy" setting. When she lands in the Sea Monster's mouth, the kinetic
energy is converted into a pulse of electricity.

What matters here is amperage, and current path. Trivially, we have the same total amount of energy at our disposal, and it could just be turned into heat by unconstrained electrical discharge through the creature; depending on the precise incidental current path, maybe that ends up mildly warming the creature (no help at all), or maybe it ends up seriously burning a small chunk, causing it great pain, and maybe actual disability or maybe not depending on exactly which small chunk gets burned.

However, again, how the energy is applied matters. If she can direct an electrical discharge to just the right part of the animal (say, through its brain, or heart), very tiny currents can cause death--you only need about 100 to 200 milliamps for less than a second to irreversibly screw up the creature's nervous system, if it's applied in just the right place.

The relevant formula in this case is $W = RI^2$, where W is power, R is resistance, and I is current. If we know how long we need the shock to last, we can replace power with total energy (which we know, from Perdita's kinetic energy), to get $E = TRI^2$. The electrical resistance of wet flesh is around 300Ω, and we want a current of around 150mA, so if we generously assume that we need the shock to last for at least 1 second to definitely incapacitate the creature, we get $E = (1s)(300Ω)(0.15A)^2 = 6.75J$. Way less than Perdita's total kinetic energy budget. So, yeah, she can electrocute the sea monster easily.

As she falls to the ground, she sees that she is headed toward a park [...]
Not wanting to become a party crashing in the most literal sense of
the term, Perdita activates the Sphere of Meteoric Conversion and
adjusts it to its "Harmless" setting. On impact, all the kinetic energy
is converted into a beam of light[....] The only damage done to the
festivities is in the form of extreme awkwardness.

Not even necessary. All of the kinetic energy can simply be dissipated as heat--into Perdita's body, the air, the ground, some thermal radiation--and if it's not specifically focused somewhere, it would hardly be noticed.

Answer 2

Well, I'm not sure if what you describe is bending the laws of physics less than transforming someone into a frog. It's a very very fancy energy converter :)

In that case, being close to physics would probably mean to assure the conservation of energy. Some examples:

• Kinetic energy: $W=\frac12mv^2$

• Electrical energy: $W=Q\cdot U=U\cdot I\cdot t$

• Heating (with heat capacity $C$): $W=C \cdot \Delta T$

• Black-body radiation : $W=P \cdot t= \sigma A T^4 \cdot t$

• LASER radiation: Found some examples for different power. Like black body radiation the power would depend on the duration of the pulse.

• Sound: $W = W_\mathrm{potential} + W_\mathrm{kinetic} = \int_V \frac{p^2}{2 \rho_0 c^2}\, \mathrm{d}V + \int_V \frac{\rho v^2}{2}\, \mathrm{d}V$, and with the sound pressure $p$ you get the sound pressure level in decibel.

Actually you would also need to consider entropy. It's always increasing of course, but does the conversion of energy require the same entropy increase than your original (kinetic) event? But then your magic probably wouldn't be possible at all. So you should accept to mess with entropy, but conserve the energy.