A particle is projected vertically upward from the origin with an adjustable initial speed u. For a fixed observation interval T, the minimum possible average speed during 0≤t≤T is:
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A particle is projected vertically upward from the origin with an adjustable initial speed u. For a fixed observation interval T, the minimum possible average speed during 0≤t≤T is:

G1 Physics · Accepted Answer

For 0≤u≤gT, the particle turns during the interval. Maximum rise: H = u²/(2g). Position at time T: y(T)=uT−gT²/2. Total distance over the interval is D = H + [H−y(T)] = u²/g − uT + gT²/2. Average speed: D/T = u²/(gT) − u + gT/2. Differentiate with respect to u: d(average speed)/du = 2u/(gT) − 1. The minimum occurs at u = gT/2. Then minimum average speed = gT/4.