The velocity is defined by:
[tex]v=\frac{dx}{dt}[/tex]
where x is the position of the particle and t is the time.
Plugging the position function given we have that the velocity is:
[tex]\begin{gathered} v=\frac{dx}{dt} \\ =\frac{d}{dt}(24t-2t^3) \\ =24-6t^2 \end{gathered}[/tex]
Hence the velocity is given by the function:
[tex]v=24-6t^2[/tex]
to determine the isntant when the velocity is zero we equate its expression to zero and solve for t:
[tex]\begin{gathered} 24-6t^2=0 \\ 6t^2=24 \\ t^2=\frac{24}{6} \\ t^2=4 \\ t=\pm\sqrt[]{4} \\ t=\pm2 \end{gathered}[/tex]
Since time is always positive we conclude that the velocity is zero at t=2 s.
Now that we know at which instant the velocity is zero we need to remember that the acceleration is defined as:
[tex]a=\frac{dv}{dt}[/tex]
then we have that:
[tex]\begin{gathered} \frac{dv}{dt}=\frac{d}{dt}(24-6t^2) \\ =-12t \end{gathered}[/tex]
hence the acceleration is:
[tex]a=-12t[/tex]
Plugging the value we found for the time we have that:
[tex]a(2)=-12(2)=-24[/tex]
Therefore the acceleration of the particle when its velocity is zero is -24 meters per second per second.
