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

JEE Main · 2026 · 23 Jan 2026 · Shift 2 · Units & Measurement

A ball of radius r and density

ρ

dropped through a viscous liquid of density

σ

and viscosity

η

attains its terminal velocity at time t, given by

t = A ρ^{a} r^{b} η^{c} σ^{d}

, where A is a constant and a, b, c and d are integers. The value of

\frac{b + c}{a + d}

____

Answer: 1

Writing

t = A ρ^{a} r^{b} η^{c} σ^{d}

dimensionally, with

[ρ] = [σ] = M L^{- 3}

[r] = L

and

[η] = M L^{- 1} T^{- 1}

, gives

[T] = M^{a + c + d} L^{- 3 a + b - c - 3 d} T^{- c}

. Matching powers of

T

- c = 1 \Rightarrow c = - 1

. Matching

M

a + c + d = 0 \Rightarrow a + d = 1

. Matching

L

- 3 a + b - c - 3 d = 0 \Rightarrow b = 3 (a + d) + c = 3 (1) + (- 1) = 2

. Hence

b + c = 2 - 1 = 1

and

a + d = 1

, so

\frac{b + c}{a + d} = \frac{1}{1} = 1

. The value is

1

JEE Main · 2026 · 23 Jan 2026 · Shift 2 · Integer answer