Two projectiles are fired with the same speed and have the same range. Their times of flight are in the ratio 1:3. If Rₘₐₓ is the maximum possible range with that speed, which pair is correct?
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Two projectiles are fired with the same speed and have the same range. Their times of flight are in the ratio 1:3. If Rₘₐₓ is the maximum possible range with that speed, which pair is correct?

G1 Physics · Accepted Answer

For equal ranges with the same speed, the launch angles are θ and 90°−θ. Since T ∝ sin(angle), T₁/T₂ = sinθ/cosθ = tanθ = 1/3. Hence tanθ=1/3. Now R/Rₘₐₓ = sin2θ = 2tanθ/(1+tan²θ) = 2(1/3)/(1+1/9) = 3/5. Maximum height is proportional to sin²θ. Therefore H₁/H₂ = sin²θ/cos²θ = tan²θ = 1/9.