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The motion of a body is given by the equation dv/dt = 6 − 3v where v is the speed in m s−1 and t is 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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The motion of a body is given by the equation dv/dt = 6 − 3v where v i...
Explanation:

Given, dv/dt = 6 − 3v

Initial condition, v=0 at t=0

We need to find the variation of speed with time.

Solution:

Separating variables,

dv/(6-3v) = dt

Integrating both sides, we get

-1/3 ln|6-3v| = t + c

Applying initial condition, v=0 at t=0, we get c = ln2

-1/3 ln|6-3v| = t + ln2

ln|6-3v| = ln2 - 3t

|6-3v| = e^(ln2-3t)

|6-3v| = e^ln2 * e^(-3t)

|6-3v| = 2 * e^(-3t)

Solving for v in both cases,

6-3v = 2e^(-3t) or 6-3v = -2e^(-3t)

v = (1 - e^(-3t)) or v = (3 + e^(-3t))

Hence, the variation of speed with time is given by

v = (1 - e^(-3t)) or v = (3 + e^(-3t))

The correct option is (a) v = (1 - e^(-3t))
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The motion of a body is given by the equation dv/dt = 6 − 3v where v is the speed in m s−1 and t is 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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