Question

The motion of a body is given by the equation dv/dt = 6 - 3v where v us the speed in m/s and t is the time in s. The body is at rest at t = 0. The speed varies with time as

(A) v = (1 - e^{-3t})
(B) v = 2(1 - e^{-3t})
(C) v = (1 + e^{-2t})
(D) v = 2(1 + e^{-2t})

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Answer 1

Given :

The motion of a body is given by the equation

• dv/dt = 6 - 3v

Body is at rest at t = 0$.$

To Find :

We have to find variation in speed with time.

Solution :

➠ dv/dt = 6 - 3v

➠ dt = dv/(6 - 3v)

Integrating both sides, we get

➠ t = -1/3 ln(6 - 3v) + C ..... (I)

where C is a constant of integration

ATQ, t = 0, v = 0

∴ 0 = -1/3 ln(6) + C

∴ C = 1/3 ln6

Putting value of C in (I), we get

➠ t = -1/3 ln(6 - 3v) + ln6

➠ t = -1/3 ln[(6 - 3v)/6]

➠ $\sf{e^{-3t}}$ = 1 - v/2

➠ v = 2(1 - $\sf{e^{-3t}}$)

∴ (B) is the correct answer.

Cheers!

Answer 2

$\large\large{\sf{\purple{\boxed{\boxed{\pink{ANSWER}}}}}}$

${\red{\longrightarrow{\boxed{\bold{v \: = \: 2(1- {e}^{ - 3t})}}}}}$