Chapter 6 - Applications of Integration - 6.1 Velocity and Net Change - 6.1 Exercises - Page 408: 32

$$s\left( t \right) = 10 + 30t - 20\ln \left| {t + 2} \right| + 20\ln 2{\text{ and }}v\left( t \right) = 30 - \frac{{20}}{{t + 2}}$$

$$\eqalign{ & a\left( t \right) = \frac{{20}}{{{{\left( {t + 2} \right)}^2}}},\,\,\,\,v\left( 0 \right) = 20,\,\,\,\,s\left( 0 \right) = 10 \cr & {\text{Use theorem 6}}{\text{.2 }} \cr & {\text{Given the acceleration }}a\left( t \right){\text{ of an object moving along a line and its initial velocity }}v\left( 0 \right){\text{, }} \cr & {\text{the velocity of the object for future time }}t \geqslant 0{\text{ is }}v\left( t \right) = v\left( 0 \right) + \int_0^t {a\left( x \right)} dx \cr & {\text{let }}a\left( x \right) = \frac{{20}}{{{{\left( {x + 2} \right)}^2}}}{\text{ and }}v\left( 0 \right) = 20.{\text{ then }}v\left( t \right){\text{ is}} \cr & v\left( t \right) = 20 + \int_0^t {\frac{{20}}{{{{\left( {x + 2} \right)}^2}}}} dx \cr & v\left( t \right) = 20 + 20\int_0^t {{{\left( {x + 2} \right)}^{ - 2}}} dx \cr & {\text{integrate}} \cr & v\left( t \right) = 20 + 20\left( {\frac{{{{\left( {x + 2} \right)}^{ - 1}}}}{{ - 1}}} \right)_0^t \cr & v\left( t \right) = 20 - 20\left( {\frac{1}{{x + 2}}} \right)_0^t \cr & {\text{evaluate the limits}} \cr & v\left( t \right) = 20 - 20\left( {\frac{1}{{t + 2}} - \frac{1}{{0 + 2}}} \right) \cr & {\text{simplify}} \cr & v\left( t \right) = 20 - \frac{{20}}{{t + 2}} + 10 \cr & v\left( t \right) = 30 - \frac{{20}}{{t + 2}} \cr & \cr & {\text{Use theorem 6}}{\text{.1 }} \cr & {\text{Given the velocity }}v\left( t \right){\text{ of an object moving along a line and its initial position }}s\left( 0 \right){\text{, }} \cr & {\text{the position function of the object for future time }}t \geqslant 0{\text{ is }}s\left( t \right) = s\left( 0 \right) + \int_0^t {v\left( x \right)} dx \cr & {\text{let }}v\left( x \right) = 30 - \frac{{20}}{{x + 2}}{\text{ and }}s\left( 0 \right) = 10.{\text{ then }}s\left( t \right){\text{ is}} \cr & s\left( t \right) = 10 + \int_0^t {\left( {30 - \frac{{20}}{{x + 2}}} \right)} dx \cr & {\text{integrate}} \cr & s\left( t \right) = 10 + \left( {30x - 20\ln \left| {x + 2} \right|} \right)_0^t \cr & {\text{evaluate the limits}} \cr & s\left( t \right) = 10 + \left( {30t - 20\ln \left| {t + 2} \right|} \right) - \left( {30\left( 0 \right) - 20\ln \left| {0 + 2} \right|} \right) \cr & {\text{simplify}} \cr & s\left( t \right) = 10 + 30t - 20\ln \left| {t + 2} \right| + 20\ln 2 \cr} $$