Question

1. A particle is moving along the x-axis so that its position at t ≥ 0 is given by s(t) = (t)In(3t). Find the acceleration of the particle when the velocity is first zero.

Answer 1

My answer choices are
3e2

e

3e

None of these

Answer 2

If \(s(t)\) describes the position, then the derivative of \(s\) gives you the velocity, i.e. \(v(t)=s'(t)\).

Find out when the velocity will be zero:
\[\begin{align*}v(t)&=s'(t)\\
&=\ln3t+t\frac{3}{3t}\\
&=\ln3t+1
\end{align*}\]
Solve for \(t\):
\[\ln3t+1=0~~\Rightarrow~~t=\frac{1}{3e}\]
The acceleration at time \(t\) is given by the derivative of the velocity function, \(a(t)=v'(t)\):
\[a(t)=\frac{3}{3t}=\frac{1}{t}\]
Velocity was found to be zero at \(t=\dfrac{1}{3e}\), so the acceleration at this time is
\[a\left(\frac{1}{3e}\right)=\frac{1}{\frac{1}{3e}}=3e\]