Question

A particle $A$ of mass $\frac{10}{7}kg$ is moving in the positive $x$ - Direction. Its initial position is $x=0$ and initial velocity is $1m/s$. The velocity (in $m/s$) at $x=10m/s$ is (use the graph given)

Answer 1

Given, mass of the particle $m=\frac{10}{7}kg$
$\text{Initial position}{x}_i=0,$
$\text{Initial velocity}=1m/s,$
$\text{Final position}{x}_f=10m,$ $\text{Final velocity}=?$
Area under the curve,
$A=\int Pdx$
$=\int m\cdot a\cdot v\cdot dx$
$\int mv\frac{dv}{dx}v\cdot dx=\int m{v}^{2}dv$

$=\frac{m}{3}[v_f^3-v_i^3]$
$A=20+\frac{1}{2}\times 2\times 10=30\text{units}$
$\frac{m}{3}[v_f^3-v_i^3]=30$
$v_f^3=\frac{90}{30}\times 7+1=64$
$v_f=4m/s$
Final Answer: $4$