Help with newtonian laws in physics problem I have this simple physics problem ( https://ibb.co/g96Lm0Q ), and I'm not quite understanding what's going on. Basically I have three forces $\vec F_v$, $\vec F_r$, and $\vec F_c$. $\vec F_v$ is constant across time. $\vec F_r$ is proportional to the speed $\vec v$ (equation is $\vec F_r= A\vec v$). $\vec F_c$ is proportional to the distance $\vec x$ from $\vec x_0$ at $t=0$ (equation is $\vec F_c= B\vec x$). A and B are constants. The object starts with a constant speed being acted upon by two equal and opposing forces $\vec F_v$ and $\vec F_r$ canceling each other out. At $t=0$, $\vec F_c$ comes into effect acting against $\vec F_v$.
One of the conditions states that across time $\vec F_v= \vec F_r + \vec F_c$. From this, it looks as if the net force never stops being zero and the speed should remain constant, yet if $\vec F_c$ is increasing (from getting further away from $\vec x=0$) this would stop being true so $\vec F_r$ must decrease.
When I do the math (see the link) it confirms that the speed of the object is in fact decreasing.
My question is how come, $\vec F_v= \vec F_r + \vec F_c$ (meaning the net force is zero and the speed should remain constant), and yet the speed is decreasing. I know I must be missing something.
 A: This condition $\vec F_v= \vec F_r + \vec F_c$ can not be correct at all times, because before $\vec F_c$ comes in,$\vec F_v=Constant$  also  velocity $\vec v$ is constant, this means $\vec F_v= \vec F_r$ (remember $\vec F_r$ is a function of $\vec v$ and is constant for now because $\vec v$ is constant).
When $\vec F_c$ comes in it changes the balance of the other two forces, so net force is not zero and the body has negative acceleration at this point. As the speed slows down, $\ F_r=A \vec v$  reduces but at the same time $\ F_c=B.x$ increases. as the body moves forward  $\ F_c=B.x$ becomes bigger and bigger, so the acceleration is negative until the body stops. So the governing equation for $\ t>0$  is  $F_v- F_r -\ F_c= m a$ $\,$ so, $\,$ $F_v- A\frac {dx}{dt} -\ Bx= m \frac {d^2x}{dt^2}$
A: The only way to keep a mass (in this case) at a constant velocity when two forces are applied on both sides of the mass is when one of the forces (in this case $F_r$) is a force caused by kinetic friction.
The force caused by $F_c$ resembles that of the force on a spring system (horizontal in this case) when stretched or compressed. Hooke's law: $F=kx$, which has the same form as in your question.
So we have a friction force $F_r$, directed to the opposite or to the same side as the spring force $F_c$ (depending on the spring being compressed or stretched). In the case of the (imaginary) spring being stretched the velocity of the mass gets smaller and so does $F_v$, but the force $F_c$ compensates for this (and vice-versa).
This implies $F_v=F_r + F_c$.  I'll leave it for you to do the math (I'm not allowed to give a complete answer).
By the way, in your differential equation, you already assumed $F_v=F_r + F_c$. Which is exactly what you have to show.
