One of the common confusion I face when writing and solving mathematical equations for physical problems is regarding signs. Consider this problem;
A ball is dropped from rest from top of the building,it falls down subjected to air resistance which is directly proportional to velocity. Find Velocity as function of time?
Let $+y$ be along downward direction
Drawing free body diagram of the body at any instant gives two forces acting on it gravity downward and air drag upwards
Assuming velocity and acceleration is downwards,writing Newton equation gives;$$mg-c(v)=m(a)$$ where $g$,$m$,$c$ are positive constants. Since this is a scalar equation since $v$ and $a$ are magnitude of the vectors. But mathematically $v$ and $a$ are variables hence can be positive or negative. Since $a={d|v|\over dt}$,the fact that speed is decreasing or increasing will mess up the sign of acceleration.
Now consider this problem;
A ball is dropped from a top of a building at a velocity greater than the terminal velocity,moving under air drag. Find $v(t)$?
Here since the ball is dropped at a velocity greater than the terminal velocity we may expect that the net acceleration is upward and velocity is downward so that it's speed decreases and it obtains terminal speed. By using same coordinate system as above and writing force equation $$mg-c(v)=m(-a)$$ but this equation is wrong in terms of sign( $(-a)$ because acceleration was in upwards direction which is negative).
So how do we deal with the sign of variables in mathematical equations in physics problems?