notes
Terminal Velocity

Terminal velocity is the maximum velocity of a body moving through a viscous fluid.

It is attained when the force of resistance of the medium is equal and opposite to the force of gravity.

As the velocity is increasing the retarding force will also increase and a stage will come when the force of gravity becomes equal to the resistance force.

After that point velocity won’t increase and this velocity is known as terminal velocity.

It is denoted by ‘v_{t}’.Where_{t}=terminal.

Mathematically:

Terminal velocity is attained when the Force of resistance = force due to gravitational attraction.
`6pieta"rv"="mg"`
`6pieta"rv"="density"xx"V"_ g "(Because density"="m"/"V")`, density = `rhosigma` where `rho` and `sigma` are the densities of the sphere and the viscous medium.
`6pieta"rv" = (rhosigma)xx4/3pir^3g` where volume of the sphere(V)=`4/3pir^3`
By simplifying
`=(rhosigma)"g"xx4/3r^2xx1/(6eta)`
`v_t=(2r^2(rhosigma)g)/(9eta)`This is the terminal velocity. Where`(v=v_t)`
Problem: The terminal velocity of a copper ball of radius 2.0 mm falling through a tank of oil at 20^{o}C is 6.5 cm s^{1}.Compute the viscosity of the oil at 20^{o}C.Density of oil is 1.5 × 10^{3} kg m^{3}, density of copper is 8.9 × 10^{3} kg m^{3}.
Solution:
Given:`"V"_t=6.5xx10^2ms^1, a=2xx10^3m,g=9.8 ms^2, rho=8.9xx10^3kgm^3, sigma=1.5xx10^3kgm^3.`
From equation:` v_t=(2r^2(rhosigma)g)/(9eta)`
`=2/9((2xx10^3m^2xx9.8ms^2)/(6.5xx10^2ms^1))xx7.4xx10^3"kgm"^3`
`=9.9xx10^1"kgm"^1"s"^1`