A particle is projected vertically upward with adjustable speed u. For a fixed observation interval 0≤t≤T, find the value of u which minimizes the RMS speed during this interval, and the corresponding minimum RMS speed.
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Question

A particle is projected vertically upward with adjustable speed u. For a fixed observation interval 0≤t≤T, find the value of u which minimizes the RMS speed during this interval, and the corresponding minimum RMS speed.

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G1 Physics · Accepted Answer

Velocity is v(t)=u−gt. The mean square speed is vᵣₘₛ² = (1/T)∫₀→T (u−gt)² dt = u² − ugT + g²T²/3. Minimize with respect to u: d(vᵣₘₛ²)/du = 2u−gT = 0. Thus u=gT/2. Substitution gives vᵣₘₛ² = g²T²/12. Therefore vᵣₘₛ,min = gT/(2√3).