The velocity of a particle

Question:

The velocity of a particle is $\mathrm{v}=\mathrm{v}_{0}+\mathrm{gt}+\mathrm{Ft}^{2}$. Its position is $\mathrm{x}=0$ at $\mathrm{t}=0$; then its displacement after time $(\mathrm{t}=1)$ is :

  1. $v_{0}+g+F$

  2. $v_{0}+\frac{g}{2}+\frac{F}{3}$

  3. $\mathrm{v}_{0}+\frac{\mathrm{g}}{2}+\mathrm{F}$

  4. $v_{0}+2 g+3 F$


Correct Option: , 2

Solution:

(2) $\mathrm{v}=\mathrm{v}_{0}+\mathrm{gt}+\mathrm{Ft}^{2}$

$\frac{\mathrm{ds}}{\mathrm{dt}}=\mathrm{v}_{0}+\mathrm{gt}+\mathrm{Ft}^{2}$

$\int \mathrm{ds}=\int_{0}^{1}\left(\mathrm{v}_{0}+\mathrm{gt}+\mathrm{Ft}^{2}\right) \mathrm{dt}$

$\mathrm{s}=\left[\mathrm{v}_{0} \mathrm{t}+\frac{\mathrm{gt}^{2}}{2}+\frac{\mathrm{Ft}^{3}}{3}\right]_{0}^{\mathrm{l}}$

$\mathrm{s}=\mathrm{v}_{0}+\frac{\mathrm{g}}{2}+\frac{\mathrm{F}}{3}$

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