A particle is projected vertically upward with adjustable initial speed u. Over a fixed time interval 0≤t≤T, u is chosen so that the maximum value of speed during the interval is as small as possible. The optimum u and minimized maximum speed are:
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Question

A particle is projected vertically upward with adjustable initial speed u. Over a fixed time interval 0≤t≤T, u is chosen so that the maximum value of speed during the interval is as small as possible. The optimum u and minimized maximum speed are:

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G1 Physics · Accepted Answer

Over the interval, v(t)=u−gt. Since v changes linearly, the largest speed magnitude occurs at an endpoint: maximum speed = max[u, |u−gT|]. To minimize the larger of these two endpoint magnitudes, make them equal: u = gT−u. Hence u = gT/2. The minimized maximum speed is then minimum possible maximum speed = gT/2.