Question

The velocity of a particle is $v=v_0+gt+F{t}^2$. Its position is $x=0$ at $t=0$; then its displacement after time $(t=1\text{s})$ is:
($g$ and $F$ are constants)

• A. $v_0+\frac{g}{2}+F$
  
• B. $v_0+2g+3F$
  
• C. $v_o+g+F$
  
• D. $v_0+\frac{g}{2}+\frac{F}{3}$
  

Answer 1

The correct option is D $v_0+\frac{g}{2}+\frac{F}{3}$


Given, $v=v_0+gt+F{t}^2$

We know that, $v=\frac{dx}{dt}$

$\Rightarrow \frac{dx}{dt}=v_0+gt+F{t}^2$

$\Rightarrow {\int }_{x=0}^{x}dx={\int }_{t=0}^{t=1}(v_0+gt+F{t}^2)dt$

$\Rightarrow x={[v_{o}t+\frac{g{t}^2}{2}+\frac{F{t}^3}{3}]}_{t=0}^{t=1}$

$\Rightarrow x=v_o+\frac{g}{2}+\frac{F}{3}$

Hence, $\left(D\right)$ is the correct answer.