A lightweight paper helicopter with rotor radius r and weight W falls through air density ρ from height h, quickly reaching terminal speed. Using dimensional analysis, find requested exponent combination in flight-time relation T = k hʳ ρᵅ rᵝ Wᵞ shown in source. [See Image for precise requested combination.]
Physics · Mixed Chapter
JEE Main Practice

Question

A lightweight paper helicopter with rotor radius r and weight W falls through air density ρ from height h, quickly reaching terminal speed. Using dimensional analysis, find requested exponent combination in flight-time relation T = k hʳ ρᵅ rᵝ Wᵞ shown in source. [See Image for precise requested combination.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Assume terminal drag scales as ρr²v², balancing weight W, so vₜ ∝ √(W/(ρr²)). Step 2 — Flight time T = h/vₜ ∝ hρ¹ᐟ² rW⁻¹ᐟ². Step 3 — Dimensional exponents in T = k hᵅρᵝrᵞWᵟ are α = 1, β = 1/2, γ = 1, δ = −1/2. Step 4 — Printed numerical answer is 1; the exact combination asked in the original PDF is missing from extractable text and requires viewing the page before assigning that value to any combination.