A ball of radius r and density dropped through a viscous liquid of density and viscosity attains its terminal velocity at time t , given by t=A ^ a r^ b ^ c ^ d , where A is a constant and a, b, c and d are integers. The value of b+c a+d is _ _ _ _ .

Step-by-step solution

Using dimensional analysis on t = A ^a r^b ^c ^d : the dimensions are [T] = [M L⁻³]^a [L]^b [M L⁻¹ T⁻¹]^c [M L⁻³]^d . Expanding: [T] = M^ a+c+d L^ -3a+b-c-3d T^ -c . Comparing powers: For T: 1 = -c , giving c = -1 . For M: 0 = a + c + d , so a + d = 1 . For L: 0 = -3a + b - c - 3d = -3a + b + 1 - 3d , giving b = 3(a+d) - 1 = 3(1) - 1 = 2 . Therefore (b+c)/(a+d) = (2-1)/1 = 1 .