$\text{v}(\text{t})=2\text{t}(3-\text{t})=6\text{t}-2\text{t}^2$
From Eq. (i) $\text{v}=6\text{t}-2\text{t}^2$$\Rightarrow\frac{\text{ds}}{\text{dt}}=6\text{t}-2\text{t}^2$
$\Rightarrow\text{ds}=(6\text{t}-2\text{t}^2)\text{dt}$
where, s is displacement$\therefore$ Distance travelled in time interval 0 to 3s,
$\text{s}=\int^3_0(6\text{t}-2\text{t}^2)\text{dt}$
$=\Big[\frac{6\text{t}^2}{2}-\frac{2\text{t}^3}{3}\Big]^3_0=\Big[3\text{t}^2-\frac{2}{3}\text{t}^3\Big]^3_0$
$=3\times9-\frac{2}{3}\times3\times3\times3$
$=27-18=9\text{m}$
$\text{Average velocity}=\frac{\text{Distance travelled}}{\text{Time}}$
$=\frac{9}{3}=3\text{m/s}$
Given, $\text{x}=6\text{t}-2\text{t}^2$$\Rightarrow3=6\text{t}-2\text{t}^2\Rightarrow2\text{t}^2-6\text{t}-3=0$
$\Rightarrow\text{t}=\frac{6\pm\sqrt{6^2-4\times2\times3}}{2\times2}=\frac{6\pm\sqrt{36-24}}{4}$
$=\frac{6\pm\sqrt{12}}{4}=\frac{3\pm2\sqrt{3}}{2}$
Considering positive sign only$\text{t}=\frac{3+2\sqrt{3}}{2}=\frac{3+2\times1.732}{2}=\frac{9}{4}\text{s}$
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