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The velocity of a particle traveling along a straight line is v = { v }_{ 0 } - ks, where k is constant. If s = 0 when t = 0, determine the position and acceleration of the particle as a function of time.

Step-by-step

Position:

\begin{aligned} (\xrightarrow { + }) \quad\quad & dt = \frac { ds } { v } \\ & \int_{ 0 }^{ t } dt = \int_{ 0 }^{ s } \frac { ds } { { v }_{ 0 } – ks } \\ & t|_{ 0 }^{ t } = -\frac { 1 } { k } \text{ ln } ({ v }_{ 0 } – ks)|_{ 0 }^{ s } \\ & t = \frac { 1 } { k } \text{ ln } (\frac { { v }_{ 0 } } { { v }_{0} – ks }) \\ & { e }^{ kt } = \frac { { v }_{ 0 } } { { v }_{ 0 } – ks } \\ & s = \frac { { v }_{ 0 } } { k } (1 – e^{ -kt }) \end{aligned}

Velocity:

v = \frac { ds } { dt } = \frac { d } { dt }[\frac { { v }_{ 0 } } { k }(1 – { e }^{ -kt })] \\ v = { v }_{ 0 } { e }^{ -kt }

Acceleration:

a = \frac { dv } { dt } = \frac { d } { dt }({ v }_{0} { e }^{ -kt }) \\ a = -k{ v }_{ 0 } { e }^{ -kt }

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