Two projectiles are fired with the same speed at complementary angles θ and 90°-θ and have common range R. Where is the vertical separation between the two trajectories maximum, and what is that maximum separation?
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Question

Two projectiles are fired with the same speed at complementary angles θ and 90°-θ and have common range R. Where is the vertical separation between the two trajectories maximum, and what is that maximum separation?

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G1 Physics · Accepted Answer

The vertical separation is Δy = x(1−x/R)|tanθ−cotθ|. The factor x(1−x/R) is a parabola with maximum at x=R/2. At x=R/2, x(1−x/R) = (R/2)(1/2) = R/4. Therefore Δyₘₐₓ = R|tanθ−cotθ|/4.

Two projectiles are fired with the same speed at complementary angles θ and 90°−θ and have common range R. Where is the vertical separation between the two trajectories maximum, and what is that maximum separation?