IGNOU MTE 14 2018 - A particle of mass m is falling under the influence of gravity through

Question

A particle of mass m is falling under the influence of gravity through a medium whose resistance equals μ times the velocity. If the particle be released from rest, show that distance fallen through time t is

$Image ignouacademy-com--p-through-29659$ $\frac{gm^2}{\mu ^2}\left \{ e^\frac{-\mu t}{m}-1+\frac{\mu t}{m} \right \}.$

15 Feb 2018

Answer :

Word Count : 507

To solve this problem, we need to apply Newton's second law of motion to the particle falling through the medium with resistance proportional to velocity. ### Step 1: Set up the equation of motion For a particle of mass $ m $ falling under gravity with resistance proportional to velocity, the forces acting on the particle are: - The gravitational force: $ F_{\text{gravity}} = mg $ - The resistive force: $ F_{\text{resistance}} = -\mu v $, where $ \mu $ is a constant and $ v $ is the velocity of the particle. By Newton's second law, the sum of forces is equal to mass times acceleration: \[ ma = mg - \mu v \] Since acceleration $ a = \frac{dv}{dt} $, we can rewrite the equation as: \[ m \frac{dv}{dt} = mg - \mu v \] Dividing both sides by $ m $: \[ \frac{dv}{dt} =

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