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Consider a small satellite of mass $m_0$ and initial velocity $v_0 >0$ that is far away from any external forces. It entered a dust cloud containing particles at rest that cling into the satellite's surface. These particles have uniform linear density $\lambda$.

Using the linear momentum conservation law, find the mass $m(x)$ of the satellite and its velocity $v(x) = \dot{x}$ after it has travelled a distance $x$ inside the cloud considering the beginning of the cloud $x=0$.

My thoughts : The mass is $m(x)= m_0+\lambda x$

Since there are no external forces we have that $$\frac{\mathrm dP}{\mathrm dt} = 0 \implies \frac{\mathrm dm(x)}{\mathrm dx}\frac{\mathrm dx}{\mathrm dt} \dot{x} + (m_0+\lambda x)\ddot{x}= 0 \implies \lambda (\dot{x})^2+m_0\ddot{x}+\lambda x\ddot{x} = 0$$ I think I can solve this ODE and find $x$ and then $v$ but there is another item on this exercise that asks me to find $x$ by integrating $v$ so I believe that there is another way of finding $v$. It has been a while since I last studied physics so I think I might be missing some detail.

Any help or hints will be appreciated

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The differential equation you have is a second-order, nonlinear equation, so it will be difficult to find its solutions directly. You can simplify it, however, by following the hint and using the chain rule to express the acceleration $a = \frac{d v}{d t} = v \frac{dv}{dx}$. This should give you a first-order differential equation that can be simply solved by separation of variables.

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