# Maximum transverse time for a partcile moving along a straight line

A particle is moving on a straight line, starts from rest, and attains a velocity $v$ after traveling a distance $d$. If the motion is such that the acceleration was never increasing, find the maximum time for the transverse.

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

$t^*=2\frac{d}{v}.$

+1

Thank you Paul. Your solution is simple and elegant.

[ answer deleted because I think I misinterpreted the statement of the problem ]

• Thank you for trying M F H. I appreciate it :)

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