Two projectiles are fired with the same speed u at complementary angles and have common range R. The sum of the areas under their trajectories is:
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Two projectiles are fired with the same speed u at complementary angles and have common range R. The sum of the areas under their trajectories is:

G1 Physics · Accepted Answer

For complementary angles, A₁ = R²tanθ/6, A₂ = R²cotθ/6. Hence A₁+A₂ = R²(tanθ+cotθ)/6. Now tanθ+cotθ = 1/(sinθcosθ). Since R = 2u²sinθcosθ/g, we have 1/(sinθcosθ) = 2u²/(gR). Therefore A₁+A₂ = R²/6 × 2u²/(gR) = Ru²/(3g).