Sequences & Series
HP
Grade 11
Question:
<p>Let \(R_1\) and \(R_2\), respectively, be the maximum ranges up and down an inclined plane and <i>R</i> be the maximum range on the horizontal plane. The \(R_1\), <i>R</i>, \(R_2\) are in</p>
<p>arithmetic-geometric progression</p>
<p>AP</p>
<p>GP</p>
<p>HP</p>
Step-by-Step Solution
Key Concept: For projectile motion on an inclined plane, the maximum ranges up and down the slope depend on the angle of inclination θ. Using R₁ = u²/(g(1+sinθ)·cos²θ) for upslope and R₂ = u²/(g(1-sinθ)·cos²θ) for downslope, their product yields R₁·R₂ = R², establishing a geometric mean relationship.
<p><strong>Step 1:</strong> For maximum range on horizontal plane: R = u²/g</p><p><strong>Step 2:</strong> Maximum range up the incline (angle θ): R₁ = u²/[g(1+sinθ)cos²θ]</p><p><strong>Step 3:</strong> Maximum range down the incline: R₂ = u²/[g(1-sinθ)cos²θ]</p><p><strong>Step 4:</strong> Calculate R₁·R₂:</p><p>R₁·R₂ = [u²/g]² · [1/(1-sin²θ)cos⁴θ] = [u²/g]² · [1/cos⁶θ·cos⁴θ]</p><p><strong>Step 5:</strong> Simplify: R₁·R₂ = (u²/g)² = R²</p><p><strong>Step 6:</strong> Therefore R₁, R, R₂ are in geometric progression with common ratio = R/R₁ = R₂/R</p><p>∴ Answer: D (Geometric Progression)</p>
Correct Answer: D