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Question: A particle moves according to the equation dv/dt = α - βv where α and β are constants. Find velocity...

A particle moves according to the equation dv/dt = α - βv where α and β are constants. Find velocity as a function of time. Assume body starts from rest.

A

v = βα(1−e−βt)\frac{\beta}{\alpha}\left( 1 - e^{- \beta t} \right)

B

v = αβ(1−e−βt)\frac{\alpha}{\beta}\left( 1 - e^{- \beta t} \right)

C

βαe−βt\frac{\beta}{\alpha}e^{- \beta t}

D

αβe−βt\frac{\alpha}{\beta}e^{- \beta t}

Answer

v = αβ(1−e−βt)\frac{\alpha}{\beta}\left( 1 - e^{- \beta t} \right)

Explanation

Solution

∫0v−βdvα− βv=−β∫0tdt\int_{0}^{v}\frac{- \beta dv}{\alpha - \ \beta v} = - \beta\int_{0}^{t}{dt}

loge (α−βv)α=−βt\frac{(\alpha - \beta v)}{\alpha} = - \beta t or v = αβ\frac{\alpha}{\beta} (1 – e-βt)