Explanation:
Given:[tex]F=m\ddot{x}=Fe^{-\frac{t}{T}}[/tex]
Solving for [tex]\ddot{x}[/tex]:
[tex]\ddot{x}=\frac{F}{m}e^{-\sqrt{\frac{F}{m} } t}[/tex]
where:
[tex]T=\sqrt{\frac{m}{F}}[/tex]
Integrating to get [tex]\dot{x}[/tex] with initial conditions [tex]\dot{x}(0)=0[/tex]:
[tex]\dot{x}=\sqrt{\frac{F}{m}}-\sqrt{\frac{F}{m}} e^{-\sqrt{\frac{F}{m}} t}[/tex]
Integrating to get x with initial conditions x(0) = 0:
[tex]x=-1+\sqrt{\frac{F}{m}} t+e^{-\sqrt{\frac{F}{m}}t}[/tex]
When t=T:
[tex]x=-1+\sqrt{\frac{F}{m}}\sqrt{\frac{m}{F}}+e^{-\sqrt{\frac{F}{m}}\sqrt{\frac{m}{F}}}=\frac{1}{e}[/tex]
[tex]\dot{x}=\sqrt{\frac{F}{m}}-\sqrt{\frac{F}{m}} e^{-\sqrt{\frac{F}{m}}\sqrt{\frac{m}{F}}}=\sqrt{\frac{F}{m}}(1-\frac{1}{e})[/tex]
