Question

Give example of a motion where x >0, u < 0 a > 0 at a particular instant

• A. $x(t)=A+B{e}^{-at}$
• B. $x(t)=A-B{e}^{-at}$
• C. $x(t)=A+B{t}^3$
• D. $x(t)=A-B{t}^3$

Answer 1

Answer: (A)

Let us consider function of motion
$x(t)=A+B{e}^{-\gamma t}$
Where $\gamma$ and $A$, is a constant $B$ is a amplitude. $x(t)$ is displacement at time $t$, where $A>B$ and $\gamma >0$
$v(t)=\frac{dx(t)}{dt}=0+(-\gamma )B{e}^{-\gamma t}=-\gamma B{e}^{-\gamma t}$
$a(t)=\frac{d}{dt}[v(t)]=\frac{d}{dt}\left(-\gamma Bex{p}^{-\gamma t}\right)=\left(\gamma B^2{\exp }^{-\gamma t}\right)$
From (i) $\therefore A>B$ so $x$ is always +ve i.e. $x>0$
From (ii) $v$ is always negative from (ii) $v<0$. From (iii) a is always again positive $a>O$.
As the value of ${\gamma }^{2}B{e}^{-\gamma t}$ can varies from o to $+\infty$.