Problem in a step of expansion of a expression.

Question

I was solving a problem in which a spaceship is accumulating dust on it when it passes by a cloud. I have a doubt that how the expansion is done, using which concept? I have marked in green box.

Answer 1

The first parenthesis is just writing $m(x+\delta x)$ in terms of $m(x)$ using equation 10.8: $$m(x+\delta x)=m_0+A\rho(x+\delta x)=(m_0+A\rho x)+A\rho\delta x=m(x)+A\rho\delta x$$ The second parenthesis contains the Taylor expansion of $v$ around $v_x$. In general $$f(x)=f(x_0)+\frac1{1!}f'(x_0)(x-x_0)+\frac1{2!}f''(x_0)(x-x_0)^2+...$$ In your case $x$ should be replaced by $x+\delta x$, and the $x_0$ in the formula is $x$ in your notation. The function $f$ is the velocity $v$. Then $$v(x+\delta x)=v(x)+\frac{dv}{dx}(x+\delta x-x)+...$$