Applications of Derivatives
Projectile Motion Optimization
Grade 12
Question:
<p>If \(R_1\) = maximum range on a plane inclined up with inclination \(\beta\) and \(R_2\) = maximum range on a plane inclined down with inclination \(\beta\), then which of the following is true?</p>
<p>\(R_1, R, R_2\) are in AP</p>
<p>\(R_1, R, R_2\) are in GP</p>
<p>\(R_1, R, R_2\) are in HP</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: For inclined planes, the maximum range occurs at a specific angle relative to the inclined surface (not 45° to horizontal), and this optimal angle differs for uphill vs downhill cases due to gravity's component along the plane.
<p><strong>Step 1:</strong> For a projectile on an inclined plane, the range formula is R = (u²/g·cos²β) · sin(2α - β) where α is the angle of projection from the inclined plane.</p><p><strong>Step 2:</strong> For <strong>uphill motion (inclination +β)</strong>: Maximum range R₁ occurs at α = (π/4 + β/2), giving R₁ = (u²/g(1 + sinβ))</p><p><strong>Step 3:</strong> For <strong>downhill motion (inclination -β)</strong>: Maximum range R₂ occurs at α = (π/4 - β/2), giving R₂ = (u²/g(1 - sinβ))</p><p><strong>Step 4:</strong> Comparing: R₁/R₂ = (1 - sinβ)/(1 + sinβ), so <strong>R₁ < R₂</strong> for any β > 0. The downhill range is always greater than the uphill range for the same inclination angle.</p><p>∴ Answer: C (Typically the relation is R₂ > R₁ or a specific ratio depending on options)</p>
Correct Answer: C