An recreational question on analysis

Question

Alice and Bob ran a marathon ($26.2$ miles) with Alice running at a uniform $8$ minutes per mile pace and Bob running erratically, but taking exactly $8$ minutes and $1$ second to complete each mile interval - that is to say all intervals of the form $[t, t + 1]$ with $t$ in $[0, 25.2]$. Can Bob have finished ahead of Alice?

Answer 1

Yes, it is possible, given that there are no limits for Bob's speed. At 208 minutes, Alice would have covered 26 miles while Bob would have covered the same in 208 minutes 26 seconds (which is still not sufficient for Alice to finish the race). The moment Bob covers 26 miles, he can accelerate and finish the race.

Answer 2

Yes. The time at which Bob uses position $x$ is given by a continuous, monotonic function $f$ and we are given that $f(x+1)-f(x)=8\frac1{60}$ for all $x\in[0,25.2]$ and of course $f(0)=0$. The question is: Can it be that $f(26.2)<8\cdot 26.2$?

Let $g\colon[0,1]\to[0,1]$ be any continuous monotonic function with $g(0)=0$, $g(1)=1$, e.g., $g(x)=x^2$. Then $$f(x)=8\tfrac1{60} \cdot \bigl(\lfloor x\rfloor +g(x-\lfloor x\rfloor)\bigr)$$ describes a possible run by Bob, and we note that he takes $$ f(26.2)=8\tfrac1{60}(26+g(0.2))=208\tfrac{13}{30}+8\tfrac1{60}g(0.2)$$ minutes to complete the run, wheras Alice takes $209\tfrac35$ minutes. So as long as $g(0.2)$ is small enough ($<\tfrac{70}{481}$, to be precise, which does hold for $g(t)=t^2$ for example), Bob can win. For the sake of physics, one may want to require the $g$ is smooth, with matching derivatives at $0$ and $1$ and with $g'$ not too small (bounded from below by the reciprocal of the speed of light) and so on, but all this doesn't forbid to find suzitable $g$ (and hence $f$).

Answer 3

Let Bob repeatedly run two tenths of a mile in one second followed by eight tenths of a mile in eight minutes. You need to verify two things: First, any one-mile stretch consists of intervals totalling two tenths of a mile where Bob is running fast and eight tenths where he's running slowly, so that his running time on any one-mile stretch totals eight minutes and one second; and second, his total running time is less than Alice's.