I am a mathematician by trade and I've stumbled upon some physics problems that I should be able to do, but just cannot. Anybody else in my predicament? Anyway.
Consider this vaguely stated problem. A mass starts moving linearly with initial velocity $v_0$, and is subjected to a friction force of $F=bxv$, $b>0$ constant, $x=x(t)$ is the distance traveled and $v(t)$ is the velocity. Find the distance it will travel and the total time of motion.
So, if I am getting this right, $v(t)=x'(t)$. I cannot try conservation of energy but Newton's second law of motion is applicable, right? So I get the ODE $$bx(t)x'(t)=mx''(t)$$ Solving this gives
$$bx^2(t)/2=mx'(t)+c \ \ \ \ \ (1)$$ Now there's my issue #1. If I let $x(0)=0$, then $c=-mv_0$. Then I get the differential equation $$mx'(t)=bx^2(t)+mv_0$$ which I can integrate to get
This seems wrong along so many lines ha,ha
How do I retrieve the total distance traveled this way? It seems like this motion will not stop.
Should I have chosen $x(0)=x_0$ so that (1) gives me a negative constant c? Then I would have a totally different $x(t)$, I think. Then how would I get the distance traveled, since it would also depend on t?