# Find velocity of small satellite [closed]

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

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.