Simple orbital collision problem -- need help!

[johnb]

I am giving my "Anime Blinded Me With Science" talk again at this year's Anime Weekend Atlanta. (The talk is on Saturday, 28 Oct, at 3:15. The convention runs Thursday 26 through Sunday 29.)

I'd like to go the extra mile on one particular bit of the talk, and I need some help with the math.

For this part, I show an opening scene from the anime "Planetes" that shows a loose aviation screw, apparently in an equatorial orbit, hitting the window of a suborbital commercial passenger spacecraft that is apparently on a polar trajectory. So, roughly traveling at right angles to each other.

In the past, I've assumed 8 km/sec for the screw, and ignored the velocity of the spacecraft.

Simple high school physics and math (KE = 0.5mv^2) gives 21.7 kilojoules for a 0.68 gram screw, 35 times more energy than a 9mm parabellum fired at point blank. So, plenty powerful enough to punch a hole in the side of the spacecraft.

Can someone help me improve the fidelity of my model?

1. Let's assume the likely apogee of a transcontinental suborbital passenger flight is 100 km. (Can we use 110 km to get more impressive numbers, or is that stretching things?) What's a simple model for the horizontal speed at apogee? (And what's the value?)

2. 100 km and V=sqrt(GM/r) gives 7.771 km/sec for the screw's horizontal velocity.

3. My calculus is super-rusty! What's the equation that will describe the relative speed at which the screw impacts the side of the spacecraft, assuming their paths are perpendicular?


I plan to use this to show kids (potentially interested in STEM careers) that:
(a) simple back-of-the envelope calculations are possible and often good enough;
(b) but there are some nifty math tools that engineers and scientists can use to really predict closely what the real values will be.


Thanks!