A ball is projected at angle α with the horizontal. For sinα

Question

A ball is projected at angle α with the horizontal. For sinα < √n/3, its distance from the projection point increases throughout flight. Find n.

G1 Physics · Accepted Answer

Step 1: Position after t is x = u cosα·t and y = u sinα·t − ½gt². Step 2: Distance squared is r² = u²t² − ug sinα·t³ + ¼g²t⁴. Step 3: dr²/dt = t[2u² − 3ug sinα·t + g²t²]. The bracket must stay nonnegative through the flight. Step 4: Its minimum is 2u² − (9/4)u²sin²α, so sin²α ≤ 8/9. Thus sinα < √8/3 and n = 8.