# Learning Goal: To practice finding the acceleration vector

Suppose an object has an initial velocity $\vec{v_{i}}$ at time $t_i$ and later, at time $t_f$, has velocity $\vec{v_{f}}$. The fact that the velocity changes tells us the object undergoes an acceleration during the time interval $\Delta \:t=t_f-t_i$. From the definition of average acceleration,

$\vec{a}=\frac{\vec{v_f}-\vec{v_i}}{t_f-t_i}=\frac{\Delta \vec{v}}{\Delta t},$

we see that the acceleration vector points in the same direction as the vector $\Delta \vec{v}$. This vector is the change in the velocity $\Delta \vec{v}=\vec{v_f}-\vec{v_i}$, so to know which way the acceleration vector points, we have to perform the vector subtraction $\vec{v_f}-\vec{v_i}$. This problem shows how to use vector subtraction to find the acceleration vector.

#### Finding the acceleration Vector

To find the acceleration as the velocity changes from $\vec{v_i}$ to $\vec{v_f}$:

1. Draw the velocity vector $\vec{v_f}$

2. Draw $\vec{-v_i}$ at the tip of $\vec{v_f}$

3. Draw $\Delta \vec{v}=\vec{v_f}-\vec{v_i}=\vec{v_f}+\left(-\vec{v_i}\right)$. This is the direction of $\vec{a}$

4. Return to the original motion diagram. Draw a vector at the middle point in the direction of $\Delta \vec{v}$; label it $\:\vec{a}$. This is the average acceleration at the midpoint between $\vec{v_i}$ and $\vec{v_f}$