# How is the direction of the instantaneous acceleration determined?

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?

• You have to know the forces or the trajectory. – jinawee Feb 22 '14 at 17:47

Take the velocity vector a two points separated by an infinitesimally small time interval. Subtract the later vector from the earlier, and divide by the infinitesimally small time interval. The resulting vector is the instantaneous acceleration (in the limit that the time interval goes to zero). The direction of that vector is the direction of the acceleration.

In finding velocity graphically, we plot the $x$ component of the position vector and the $y$ component of position vector for each instant of time. We end up with a curve: the trajectory. The direction of the velocity is tangent to the curve at each point of time. The analog of that method to find the direction of acceleration is to plot $v_y$ and $v_x$ for each instant of time on graph paper. The direction of acceleration is tangent to that curve at each point of time.

You can't find the direction of acceleration directly from the trajectory, although if you had some extra information, namely the value of time at each point on the trajectory, you could figure it out with some work.

• by definition the direction of the average acceleration is the same as that of delta v. so not only the magnitude but the direction too can be find by using the concept of limit. – KawaiKx Feb 23 '14 at 13:16

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.