A simple differential equation question.

[MathematicalPhysicist]

problem statement
A satellite travels in interstellar space, in its motion it collects stardust and its mass increases by $$\frac{dM}{dt}=\rho Av$$
A is the surface area of the satellite, rho is the stardust density, and v is the the speed of the satellite, at time t=0 the satelite speed is v0,and mass M0, you may assume that no external forces are acting upon the system, find the speed as function of time.

attempt at solution
well it looks simple $$0=\frac{dp}{dt}=\frac{dM}{dt}v+\frac{dv}{dt}M(t)$$
where $$M(t)=\rho Ax+M0$$ where x is the displacement the satelite goes, which yields: $$0=\rho A(\frac{dx}{dt})^2+\frac{d^2x}{dt^2}M(t)$$ here I am kind of stuck, if i ofcourse got it correct with the equation.
i think i can solve it with lowering the order of the equation, but first i want to see if i got the physics correct, so have i?

thanks in advance.

 

[muppet]

Looks ok to me! :)

 

[m2003]

Dear writer,

Your attempt at the solution is on the right track, but there are a few things that can be improved. First, the equation you are using is not a differential equation, it is a kinematic equation for the motion of an object. In order to solve for the speed as a function of time, we need to use the equation you provided in the problem statement, \frac{dM}{dt}=\rho Av. This is a differential equation because it involves the derivative of the mass with respect to time.

To solve this differential equation, we can use the technique of separation of variables. We can rewrite the equation as \frac{dM}{\rho A}=vdt. Then, we can integrate both sides to get M=\rho Avt + C, where C is a constant of integration. We can use the initial conditions given in the problem, t=0 and M=M0, to solve for C. This gives us the equation M=\rho Avt + M0.

Now, we can use this equation to solve for the speed as a function of time. We know that speed is equal to the change in distance over the change in time, or v=\frac{dx}{dt}. We can use the equation for M that we just found to substitute for t and M, giving us v=\frac{dx}{dt}=\frac{d}{dt}(\rho Avt+M0)=\rho Av. This means that the speed is directly proportional to the stardust density and surface area of the satellite.

In summary, your physics is correct, but the approach to solving the problem needs to be adjusted. Keep in mind that differential equations require a different method of solving compared to kinematic equations. I hope this helps!