Question

A particle is moving along the x-axis so that its position at t ≥ 0 is given by s(t) = (t)In(5t). Find the acceleration of the particle when the velocity is first zero. (4 points)

5e

5e2

e

None of these

Answer 1

The particle's velocity is given by the derivative of its position:

$s'(t)=\ln5t+t\dfrac5{5t}=\ln5t+1$

which will be 0 when

$\ln5t+1=0\implies\ln5t=-1\implies5t=e^{-1}\implies t=\dfrac1{5e}$

The acceleration is given by the second derivative, so we have

$s''(t)=\dfrac5{5t}=\dfrac1t$

and at $t=\dfrac1{5e}$, the acceleration will be

$s''\left(\dfrac1{5e}\right)=\dfrac1{\frac1{5e}}=5e$