Question

A point initially at rest moves along $x$ -axis. Its acceleration varies with time as $a=(6 t+5) \,m / s ^{2} .$ If it starts from origin, the distance covered in $2 s$ is

• A. 20 m
• B. 18 m
• C. 16 m
• D. 25 m

Answer 1

Answer: (B)

18 m

Answer 2

Given, $a=\frac{d v}{d t}=6 t+5$
or $d v=(6 t+5) d t$
Integrating, we get
$\int\limits_{0}^{v} d v=\int\limits_{0}^{t}(6 t+5) d t$
or $v=\left(\frac{6 t^{2}}{2}+5 t\right)$
Again $v=\frac{d s}{d t}$
$\therefore d s=\left(\frac{6 t^{2}}{2}+5 t\right) d t$
Integrating again, we get
$\int\limits_{0}^{s} d s=\int\limits_{0}^{t}\left(\frac{6 t^{2}}{2}+5 t\right) d t$
$\therefore s=\frac{3 t^{3}}{3}+\frac{5 t^{2}}{2}$
when, $t=2 s, s=3 \times \frac{2^{3}}{3}+\frac{5 \times 2^{2}}{2}$
$=3 \times \frac{8}{3}+\frac{5 \times 4}{2}$
$=8+10=18\, m$