Fast pennies

[Stefan Ram]

What would happen if a penny with a mass of 0.003 kg and a speed of

0.99999999999999999999999999999999999999 c

from outer space would hit the earth (being directed at its center)?


[[Mod. note --
There are 38 9's in that speed, i.e., the speed is (1 - 1e-38)*c.
That implies a Lorentz gamma factor of (1 - v^2/c^2)^(-1/2) = 7e18,
so the penny's total relativistic energy is gamma*m*c^2 = 2e33 Joules.
That's rather a lot of energy. :) In fact, it's about 8 times the
Earth's gravitational binding energy (2.5e32 Joules according to the
all-knowing Wikipedia).

So, the tricky question is, how much of the penny's energy would go
into disrupting the Earth, versus how much would go into kinetic energy
of whatever came out the other side?

And finally, I'll note that (IMHO not superb, but still an enjoyable read)
the 1993 science-fiction novel "Flying to Valhalla", by Charles Pellegrino,
is based on a similar question.
-- jt]]

[Richard Livingston]

Post by Stefan Ram
What would happen if a penny with a mass of 0.003 kg and a speed of
0.99999999999999999999999999999999999999 c
from outer space would hit the earth (being directed at its center)?

You might want to look at:

https://what-if.xkcd.com/1/

This is a reasonably good analysis of a baseball going only 0.9c. Your
penny would have a couple orders of magnitude greater energy, but
the effect would be pretty much the same, but greater scale.

Rich L.

[[Mod. note -- It's actually rather more than "a couple orders of
magnitude": the total energy (gamma*m*c^2) is around 10^17 times
larger for the penny (2e33 Joules) than for the baseball (3e16 Joules).

Stefan Ram's penny would have (much) more than enough energy to
vaporize the entire Earth.
-- jt]]

[wugi]

Post by Stefan Ram
What would happen if a penny with a mass of 0.003 kg and a speed of
0.99999999999999999999999999999999999999 c
from outer space would hit the earth (being directed at its center)?
[[Mod. note --
There are 38 9's in that speed, i.e., the speed is (1 - 1e-38)*c.
That implies a Lorentz gamma factor of (1 - v^2/c^2)^(-1/2) = 7e18,
so the penny's total relativistic energy is gamma*m*c^2 = 2e33 Joules.
That's rather a lot of energy. :) In fact, it's about 8 times the
Earth's gravitational binding energy (2.5e32 Joules according to the
all-knowing Wikipedia).
So, the tricky question is, how much of the penny's energy would go
into disrupting the Earth, versus how much would go into kinetic energy
of whatever came out the other side?
And finally, I'll note that (IMHO not superb, but still an enjoyable read)
the 1993 science-fiction novel "Flying to Valhalla", by Charles Pellegrino,
is based on a similar question.
-- jt]]

Wouldn't it pass almost unnoticed through the Earth? The reaction time
with whatever obstacles it encounters would exceed largely its time of
passing by, so there would be hardly "explosive information" passed on
to them, or would there?

[Stefan Ram]

Post by Stefan Ram
so the penny's total relativistic energy is gamma*m*c^2 = 2e33 Joules.
That's rather a lot of energy. :)

I came up with this value by searching for an energy that is
somehow in an order of magnitude that could "shake the earth"
and then came up with the kinetic energy that the earth has due
to its mass and orbital velocity. The idea was that it would
"shake the earth" if the earth hit a stationary obstacle.

Post by Stefan Ram
There are 38 9's in that speed, i.e., the speed is (1 - 1e-38)*c.

There is a medium that could give the earth some protection:
The cosmic background radiation. From the perspective of the
fast penny, photons of the background radiation could become
so energetic that they slow down and/or destroy the penny.

A penny that fast would have a lot of momentum too. Transfer of
some of that momentum could change the earth's orbit.

Some concepts that might be relevant in this regard are the
cross sections of and the energy transferred by objects at
a certain speed, and in this regard one can sometimes hear
"Froissart theorem" and "stopping power".

|The Froissart theorem (or Froissart bound) is known since
|1961, after publication of the paper [1]. Its main statement
|says that the total cross section of two-hadron interaction
|cannot grow with energy faster than log2 E.
...
|[1] M. Froissart, Phys. Rev. 123 (1961) 1053.

(So, one can get the impression from reading this that the
cross section can /grow/ with energy.)

|In nuclear and materials physics, stopping power is the
|retarding force acting on charged particles, typically alpha
|and beta particles, due to interaction with matter, resulting
|in loss of particle kinetic energy. Stopping power is also
|interpreted as the rate at which a material absorbs the
|kinetic energy of a charged particle. Its application is
|important in a wide range of thermodynamic areas such as
|radiation protection, ion implantation and nuclear medicine.
...
|The deposited energy can be obtained by integrating the
|stopping power over the entire path length of the ion while it
|moves in the material.

Another word I found helpful for searches is "ultrarelativistic".