The Running and Throwing Up equation

Question

About a month ago, I was at an outdoor university function and I was taking full advantage of the ample supply of free food and beer being offered. Not less than an hour into this event, I spotted a friend of mine, whom I hadn't seen in a while. At this point in time, my stomach was practically saturated in all the meat, beer, and cheese I had been consuming, and I knew that running after her would land me in the demise of time's reappropriation of my dinner. So as I stood there, I thought to myself: "For every moment I wait, I must run a longer distance, yet I simultaneously reduce my chances of vomiting while running." I figured the analytical discription of this scenario would involve a few factors: vomit probability (dependent on time and speed), speed itself (with exhaustion preventing me from running constantly), distance and changing of distance to the person, and lastly time. I suppose the question I wish to ask is: given an increasing change in distance from my destination, how much time must I let pass to optimize my chances of catching up with my friend? I realize this requires an explicit probability for vomiting and equations for speed and destination. Though if anyone could help me get a head start, I'd greatly appreciate it.

Answer 1

I think that the first step would be to model the problem. You have found the relevant parameters (velocity of your friend walking $v_{friend}$, your own walking speed $v$, the time you start walking $t_0$, the distance between you and your friend $d_{friend}=d_{0, friend}+v_{friend}t-v(t-t_0)$, time, your probability to vomit $P(v,t,t_0)$...)

First if you want to reach your friend, you need to have supposing $v$ and $v_{friend}$ constant :

$$v>v_{friend}$$