A freely falling particle crosses two equal windows, each of height h. It crosses the upper window in time 2t and the lower window in time t. The vertical separation between their top edges is:
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A freely falling particle crosses two equal windows, each of height h. It crosses the upper window in time 2t and the lower window in time t. The vertical separation between their top edges is:

G1 Physics · Accepted Answer

Let v₁ be the speed at the top of the upper window. For the upper window: h = v₁(2t) + (1/2)g(2t)² ⇒ v₁ = h/(2t) − gt. Let v₂ be the speed at the top of the lower window. For the lower window: h = v₂t + gt²/2 ⇒ v₂ = h/t − gt/2. If S is the vertical separation between the two top edges, then v₂² = v₁² + 2gS. Therefore S = [v₂²−v₁²]/(2g) = 3[h²/t² − g²t²]/(8g).