bottomless pit: ultimate bungee

So a radio host’s silly quiz game had the failing caller “fall into a bottomless chasm” or some such phrasing. That got me thinking: it’s bottomless, so there is nothing downwards from it; but if it’s truly bottomless, you’d go through the center of the earth, at which point you could hit a “ceiling” at a rather high speed, since you’d be going upwards at this point. But if there were no ceiling, that would be like ultimate bungee!

Okay, it really wouldn’t. Dr. Christopher Baird’s blog has a detailed analysis, which points out (as I put it) “if the pressure doesn’t kill you, the heat will”. With atmosphere above you (on the way down), pressure will increase until you cannot breathe (20 atm of pressure at 0.15km), just like a submarine deep enough in the ocean. And it gets really, really hot the deeper you go (320 Kelvin at 1.1km will kill you of heat stroke; 1200 Kelvin by 200km will incinerate what’s left).

But assuming you could protect yourself from the pressure and temperature (magic elevator cage), and assuming you liked bungee, that would be fun. Well, not bungee, yet, because the atmosphere, aside from crushing you, will decelerate you, too; you’re stuck at terminal velocity (120mph or 200kph for a human-shaped elevator). So assuming you evacuated the pit, aligned the ends so they both punch through land rather than ocean, and could withstand the temperature, that would be a super-speed bungee.

Back in May 2014, I made a spreadsheet to make nice potential energy graphs:

potential energy

F(r) = -G M m / r^2		
U = INTEG{INF to r}[ F(r) dr ]

r >e; Re
	M(r>Re) = Me		
	F(r>Re) = -G Me m / r^2		
	U(r>e;Re) = -G Me m / |r|

r <e; Re
	M(r<Re) = rho * (4/3) * pi * r^3		
	F(r<Re) = -G M(r) m / r^2		
	U(r<Re) = U(r=Re) - G m rho * (4/3) * pi * (Re^2 - r^2)/2

Then, find U(r=Re) and U(r=0): the delta will be the kinetic energy gain at the center of the earth. With K=1/2*m*v^2, I find that at the center, going about 8km/sec (almost 18000mph)!

Also, my old physics book (Halliday & Resnick (2nd ed), pg251) showed the calculation for how long it takes to do the round trip: because inside the uniform-density sphere approximation of the earth, force is directly proportional to distance from the center [F(inside)=-kx], then simple harmonic motion applies, and the period can be easily calculated at 84min, so you’d make the round trip in under 1.5hours.

But, once again, your radio host would have to be rather accomodating, especially if you want him to let you out of the elevator cage at the end of the trip. 🙂

(GSU’s HyperPhysics also shows the period calculation. And Wikipedia:Gravity_train brings up some of the results and issues involved.)