- #1

MxwllsPersuasns

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## Homework Statement

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This is a problem from my Differential Geometry course

A velociraptor is spotting you and goes after you. There is a shelter in the direction perpendicular to the line between you and the raptor when he spots you. So you run in the direction of the shelter at a constant speed v. The raptor is pursuing you (also at a constant speed w) adjusting at each moment in time t his direction to the line between you and him (pursue in the direction of the line of sight). Calculate the curve γ(t) the raptor is following. Do some calculations (taking realistic speeds into account) whether you would make it to your shelter (play with the distance to the shelter compared to your and the raptor’s running speeds). Draw a picture of the situation.

## Homework Equations

## The Attempt at a Solution

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So I basically set this problem up as follows:

I made the line between myself and the creature the y-axis (vertical axis) with the distance between us initially at d. Then the line from our midpoint to the shelter was the horizontal (t) axis. I noticed my path to the shelter made a right triangle (looks like a 30-60-90 one) from the upper endpoint of the y-axis to the shelter which was (t = ζ) units along the t axis. Now if the velociraptor adjusts his direction to pursue me at every instant his graph will look something like the Ln(t) graph.

So I continued with the analysis by determining the lengths of each corresponding path (my linear path and the beasts logarithmic path) of which I found...

- Mine was {(d/2)

^{2}+ ζ

^{2}}

- The killers was (after the arc length formula) √[1/(t^2)]*t*ln(t) which ran from 0 to ζ

* However because of the 0 endpoint I kept getting ∞ as an answer for the length (presumably due to the asymptotic behavior of the logarithmic function at t=0) so to try and rememdy this I set the lower bounds to be an absurdly small number (in terms of realistic timing) and set it as 10^-10. When I ended up calculating the lengths and times I ended up getting answers that were way off for the raptor (the human seemed to be fine)

So I guess my question is what way would you guys approach this or can you help me see the faults in my argument? Because the way I set it up the only real curve that I could see describing the path the raptor takes is ln(t) but when I try to use that at t=0 I run into issues. Any help is incredibly appreciated..