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A particle is moving along a straight line such that its acceleration is defined as a = (-2v) m/{ s }^{ 2 }, where v is in meters per second. If v = 20 m/s when s = 0 and t = 0, determine the particle’s position, velocity, and acceleration as functions of time.

Step-by-step

a = -2v \\ \frac { dv } { dt } = -2v \\ \int_{ 20 }^{ v } { \frac { dv } { v } } = \int_{ 0 }^{ t } { -2\space dt } \\ \text{ ln } \frac { v } { 20 } = -2t \\ v = ( { 20e }^{ -2t } ) \text{ m/s } \\ a = \frac { dv } { dt } = ({ -40e }^{ -2t }) m/{ s }^{ 2 } \\ \int_{ 0 }^{ s } { ds } = v\space dt = \int_{ 0 }^{ t } { ({ 20e }^{ -2t })dt } \\ s = { -10e }^{ -2t }|_{ 0 }^{ t } = -10({ e }^{ -2t } – 1) \\ s = 10(1 – { e }^{ -2t })\text{ m }

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