Maximum transverse time for a partcile moving along a straight line
2 Answers
Let $t^*$ be the maximum time for the transverse.
Think about the graph of the vecolicty $v=v(t)$. We know that $v(0)=0$, and the graph is never concave up, since $\frac{dv}{dt}$ is never incresing. The area under the graph of $v(t)$ is equal to the traveled distance $d$, i.e. $$d=\int_0^{t^*} v(t)dt.$$
It is clrear that the transverse time will be maximized when the curve $v(t)$ from $0$ to $v$ is a straight line. At the maximum time $t^*$ we have
\[\frac{1}{2}t^* v=d.\]
Hence the maximum time for the transverse is
\[t^*=2\frac{d}{v}.\]
Paul F
200
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Thank you Paul. Your solution is simple and elegant.
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