[Physics] How is the direction of the instantaneous acceleration determined

Question

I know from the text book that the direction of velocity at any point on the 2D path of an object is tangential to the path at that point and is in the direction of motion. But how would one determine the direction of instantaneous acceleration? In case, the only force influencing is gravity, direction of instantaneous acceleration is downward towards the earth's center. but what in the other cases?

Answer 1

For some vector function $\vec {v}(t)$ that gives the velocity:

$\vec{a}(t) = \dot{\vec{v}}(t)$

If velocity is not given as a vector function, then you need additional information to find the direction of the acceleration.