A particle is moving along the x-axis so that its position at t ≥ 0 is given by s(t) = (t)In(2t). Find the acceleration of the particle when the velocity is first zero.

Question

solomonzelman · Answer

$\color{black}{\rm Example}$:
A particle is moving along the $\color{black}{\displaystyle x}$-axis so that its position  (in meters)
at $\color{black}{\displaystyle t\ge0 }$ is given by $\color{black}{\displaystyle s(t)=t^4-12t }$  (in seconds).  Find the acceleration
of the particle when the velocity is 20m/s.


$\color{black}{\rm Solution}$:
$\color{black}{\displaystyle s(t)=t^4-12t }$
$\color{black}{\displaystyle s'(t)=4t^3-12}$     (or $\color{black}{\displaystyle v(t)=4t^3-12}$).

When (at what time) is the velocity 8m/s?
$\color{black}{\displaystyle v(t)=20}$
$\color{black}{\displaystyle 4t^3-12=20}$
$\color{black}{\displaystyle 4t^3=32}$
$\color{black}{\displaystyle t^3=8}$
$\color{black}{\displaystyle t=2}$.

What is the function for acceleration?
$\color{black}{\displaystyle s'(t)=4t^3-12}$
$\color{black}{\displaystyle s''(t)=12t^2}$     (or $\color{black}{\displaystyle a(t)=12t^2}$).

So, at this time (at t=2) what is the acceleration?
$\color{black}{\displaystyle a(2)=12(2)^2=48}$   (m/s$^2$)

solomonzelman · Answer

Correction:  When (at what time) is the velocity $\color{red}{20}$m/s?     $\small \color{black}{(20 ~~{\rm instead~~of~~} 8)}$