A ball moves successively through three liquids, at rest as shown, of densities `sigma_(1),sigma_(2)` and `sigma_(3)` and viscosity coefficient `eta_(1),eta_(2)` and `eta_(3)` and respectively with the same terminal velocity then

Options

A. `eta_(3)gteta_(2)gteta_(1)`
B. `(sigma_(1))/(eta_(1))=(sigma_(2))/(eta_(2))=(sigma_(3))/(eta_(3))`
C. `(eta_(1))/(eta_(3)gt(eta_(3))/(eta_(2))`
D. `(eta_(2)sigma_(1)-eta_(1)sigma_(2))/(eta_(3)sigma_(1)-eta_(1)sigma_(3))=(eta_(2)-eta_(1))/(eta_(3)-eta_(1))`

Answer

C, D

Explanation

`(rho-sigma_(1))/(eta_(1))=(rho-sigma_(2))/(eta_(2))=(rho-sigma_(3))/(eta_(3))`
`sigma_(1)ltsigma_(2)ltsigma_(3)implieseta_(1)gteta_(2)gteta_(3)`
Also, `(eta_(1))/(eta_(3))=(rho-sigma_(1))/(rho-sigma_(3))gt1(eta_(3))/(eta_(2))=(rho-sigma_(3))/(rho-sigma_(2))lt1implies(C)`
Eliminating `rho` gives (D)