In Atwood machines, is the direction of the acceleration an observational fact or can be deduced mathematically? In Atwood machines type problems, we set one block heavier than the other. Then we usually "know" the direction of rotation of the pulley and the direction of acceleration based on the location of the heaviest block: If the heaviest block is on the right side, the acceleration is clockwise, if the heaviest is on the left, counter-clockwise.
Is this an experimental fact based on observation or can be deduced from a mathematical perspective?
 A: You don't have to know which block is heavier than the other. But you do have to adopt some sign convention for the direction of the acceleration.
Without actually knowing which mass is heavier, you can arbitrarily assign the acceleration $a$ is to be positive for say mass 1 moving downward, and then applying Newton's second law you get the equation for the acceleration to be
$$a=g\frac{m_{1}-m_{2}}{m_{1}+m_{2}}$$
Now, if in fact you are given that $m_{2}>m_{1}$ then $a$ will be negative, and $m_{1}$ accelerates upward instead of downward.
But you could just as well assign $a$ to be positive for $m_{2}$ moving downward, in which case the equation is.
$$a=g\frac{m_{2}-m_{1}}{m_{1}+m_{2}}$$.
With the same result.
It's kind of like the application of Kirchhoff's voltage law where you arbitrarily assign a direction to a loop current. If as a result of analysis that current has a negative value, then the current is in the opposite direction to that assumed.
Hope this helps.
A: With two masses hanging over a pulley, Newton's laws predict that the heavier mass will determine the direction of motion.
