Okay, I’ve got some free time, I’m going to science the shit out of this!
How many cheeseburgers do I need to cast Feather Fall?
(Disclaimer: I am a mathematician, not a physicist. If I use a formula wrong somewhere, feel free to let me know and we can politely debate science)
We have a friend, let’s call him Mike (sorry anyone reading this who is named Mike) who is falling at terminal velocity. We’re assuming gravity is the same as on earth (because we’re lazy) and we’re rounding it to 10 m/s² (because we’re lazy- it’s actually 9.807 m/s² but that would mean i need to go get out a calculator so 10 it is!).
To get the force that Mike is experiencing, we need to multiply his acceleration (10 m/s²) by his mass (which we’re assuming is about 70 kg, the average mass of a person). So the force he’s experiencing is 10*70 = 700 Newtons of force on Mike. We need to counter this force if we want Mike to stop falling towards the ground. We also ideally want to slow him down enough so that he is not falling at terminal velocity when he hits the ground, because that would kill him. Terminal velocity on earth is about 54 m/s; with a parachute this goes down to about 7-8 m/s, which is what we’re aiming for. If we decelerate at 10 m/s² we can achieve this in about 5 seconds. A human can withstand acceleration of about 49 m/s²; bear in mind that we’ve got to take existing gravity into account. We could decelerate at 39 m/s², which slows us down in a couple of seconds, but that is going to push the limits of what Mike can withstand. It depends how close he is to the ground as to how fast we want to decelerate- if he’s too close, we might not have the time to slow him down fast enough without killing him anyway.
We’re going to assume he’s about 500 metres up in the air; he’s high up enough that we’ve got time to slow him down, but he’s going to hit the ground in about 10 seconds if we don’t do something about it. Time for some maths!
How much energy does it take to stop gravity?
Now, we want to know the potential energy related to Mike’s current fall, because this will tell us how much energy we need to counteract this. He’s currently got 700 Newtons of force pulling him down; this means that for every metre he falls, it releases 700 Joules of potential energy. This means that, to stop the force of gravity from making him accelerate any faster, we need to send 700 Joules of energy to push him in the other direction- and this is the amount we need to send every second while he’s up there, because gravity does not have an off switch.
So, 700 Joules per second to start with. Air resistance is going to deal with this until we start slowing his fall, but once we start slowing him down we are going to need to keep up the 700 Joules per second until he hits the ground, otherwise he will start to accelerate again. That’s already going to be quite a lot of energy; we certainly don’t want to leave Mike suspended in midair for a long time.
How much energy do we need to slow his fall?
So, the 700 Joules per second means that Mike is no longer moving any faster towards the earth. We still need to deal with getting our speed of 54 m/s down to 7 m/s so that Mike won’t die when he hits the ground.
If we decelerate at 10 m/s² for 5 seconds (bringing us down to 4 m/s because a. slower is better and b. I’m too lazy to fetch the exact number of seconds we need) (ignoring what we’ve already done to take out the effects of gravity), we will use 3500 Newtons of force. In that time we will fall (54^2 - 4^2)/2*10 = (with the help of a calculator)
145 metres. Then we can calculate the work done, which is force * distance = 3500 * 145 = 507500 Joules.
So, we have an initial output of energy to slow down Mike, for 507500 Joules, and we need a good 145 metres of space before he hits the ground to do this. Then while we’re doing that we need to expend 700 Joules per second to counteract the effects of gravity, which will be happening for at least 5 seconds while he’s decelerating, possibly longer once he’s slowed down and not at the ground yet.
Because Mike is our friend and we’re very conscious about the fact that we don’t want to accidentally wait too long before we slow the fella down, we’re going to start slowing Mike down now. We have an initial output of 507500 Joules to slow him down, plus another 700 * 5 = 3500 Joules for counteracting gravity while he decelerates. that’s 510100 Joules so far for anyone keeping score at home.
So Mike is now floating down at a leisurely 4 metres per second. He’s still 500-145 = 355 metres up in the air, so he will be travelling down for another 89 or so seconds. That’s another 62,300 Joules of energy to stop gravity while he’s decelerating, making our grand total to be 572400 Joules to get Mike safely to the ground.
Where are we getting this energy?
We’ve been working in Joules, but most foods are defined in KiloJoules, so let’s convert: we have to find 572.4 kJ to stop Mike from hitting the ground. For context: a cheeseburger has about 3000 kJ, so we can reliably save 5 Mikes on a happy meal (assuming we eat something as well- your body needs fuel too!) (this also says something about how insanely weak the force of gravity is).
Of course, this essay doesn’t take into account what energy requirements you need to project energy, or send energy that far, but here’s a comforting thought; Mike is going to live, thanks to you!
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