Question Detail

Force has dimensions [M L T⁻²]. Density ρ has dimensions [M L⁻³]. Velocity v has dimensions [L T⁻¹]. Area a has dimensions [L²]. We need to combine a, v, and ρ to get [M L T⁻²]. Let F ∝ a^x v^y ρ^z. Then [L²]^x [L T⁻¹]^y [M L⁻³]^z = M^z L^(2x + y - 3z) T^(-y). Equating: M: z = 1. T: -y = -2 => y = 2. L: 2x + y - 3z = 2x + 2 - 3 = 2x -1 = 1 => 2x = 2 => x = 1. So F ∝ a¹ v² ρ¹. Thus dimensions are [a v² ρ].