question_answer 46)

                  A particle executes the motion described by                 \[x(t)={{x}_{0}}(1-{{e}^{-\gamma t}});\,\,t\ge 0\].                 (a) Where does the particle start and with what velocity?                 (b) Find maximum and minimum values of \[x(t),\,\upsilon (t),\,\,a\,(t)\]. Show that \[x(t)\] and \[a\,(t)\] increases with time and \[\upsilon (t)\] decreases with time.                

Answer:

                  \[x(t)\,=\,{{x}_{0}}\,(1\,-{{e}^{-\gamma t}});\] \[\upsilon (t)\,=y\,{{k}_{0}}{{e}^{-\gamma t}}\] and \[a(t)\,=-\,{{y}^{2\,}}{{x}_{0}}{{e}^{-yt}}\]                 (a) When \[t\,=0,\,\,x=\,{{x}_{0}}\,(1\,-\,{{e}^{-0}})=\,0\] and \[\upsilon \,=\,y{{x}_{0}}\]                 (b) \[x\,(t)\,|{{\,}_{\max }}\,={{x}_{0,}}\] \[\upsilon \,\,(t)\,|{{\,}_{\max }}\,=y{{x}_{0}}\] \[a(t)\,|{{\,}_{\max }}\,={{y}^{2}}{{x}_{0}}\]