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Grade: 12

                        

What is the time taken to attain 0.63 times terminal velocity by a body of mass m and radius r moving through a liquid of density rho Please help

2 years ago

Answers : (1)

Arun
24742 Points
							
When an object moves through a viscous medium such as air or water, the object experiences a velocity dependent retarding force. This retarding force may be proportional to the first (linear) or the second power (quadratic) of the velocity (a case often studied in classical mechanics).

We will model the movement of an object moving through a viscous medium with the following differential equation:

m.dV/dt = mg - kV

where m is the mass of the object,

k is a scaling or proportionality factor that accounts for the area experiencing the viscous force, dimensional formula is MT-1

V is the velocity,

=> integral of [dV/(mg - kV)] limits from 1 to 0.63Vt = integral of [ dt/m ] limits from 0 to T(time constant)

on solving gives T=(m/k)*ln[mg / (mg - 0.63Vt)]

where its dimesnion formula  is T

2 years ago
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