If the solve the problem

Question:

If $s=t^{3}-4 t^{2}+5$ describes the motion of a particle, then its velocity when the acceleration vanishes, is

(a) $\frac{16}{9}$ unit / sec

(b) $-\frac{32}{3}$ unit/sec

(c) $\frac{4}{3}$ unit/sec

(d) $-\frac{16}{3}$ unit/sec

Solution:

(d) $-\frac{16}{3}$ unit / sec

According to the question,

$s=t^{3}-4 t^{2}+5$

$\Rightarrow \frac{d s}{d t}=3 t^{2}-8 t$

$\Rightarrow \frac{d^{2} s}{d t^{2}}=6 t-8$

$\Rightarrow 6 t-8=0$       $\left[\right.$ As velocity deminishes, then $\left.\frac{d^{2} s}{d t^{2}}=0\right]$

$\Rightarrow t=\frac{4}{3}$

Now, $\left(\frac{d s}{d t}\right)_{t=\frac{4}{3}}=3\left(\frac{4}{3}\right)^{2}-8\left(\frac{4}{3}\right)$

$\Rightarrow \frac{d s}{d t}=\frac{16}{3}-\frac{32}{3}$

$\Rightarrow \frac{d s}{d t}=-\frac{16}{3}$ unit $/ \mathrm{sec}$

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