Two solild spheres manufactured of the same material freely fall down in the air. One sphere has a dimeter twice as large as the other. The force due to air resistance is proportional to the cross-section area of a moving object and is quadration function of the speed of an object and is quadration function of the speed of an object. In sometime after the beginning of motion in the presence of air resistance, the velocity of each sphere become constant. It is called the terminal velocity `v_(big)//v_(small)` is

Options

A. `(1)/(sqrt(2))`
B. `(1)/(2)`
C. `sqrt(2)`
D. `2`

Answer

C

Explanation

An object will attain terminal velocity when force of gravity on object is equal to the force of the air resistance, i.e., Mg `=k v^(2) A`
or `(4)/(3)pi r^(3)rho g = kv^(2)pi r^(2)` …..(i)
where M is the mass of the sphere, r is its radius and `rho` is the density of the material of sphere and k is a constant of proportionality.
From (i), `v prop sqrt(r)`
`(v_(big))/(v_(small)) =sqrt((r_(big))/(r_(small))) =sqrt((2r)/(r)) =sqrt(2)`