Average and Instantaneous Velocities

To describe an object's position in a plane, a single coordinate is no longer enough — both an x-coordinate and a y-coordinate (or, equivalently, one position vector $\vec{r}$) are needed. Suppose the object is at point P at time $t_1$, with position vector $\vec{r}_1 = x_1\hat{i} + y_1\hat{j}$, and has moved to point Q at time $t_2$, with position vector $\vec{r}_2 = x_2\hat{i} + y_2\hat{j}$.

The displacement of the particle from $t_1$ to $t_2$, shown as the arrow PQ, is $\Delta \vec{r} = \vec{r}_2 - \vec{r}_1 = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j}$, exactly the two-dimensional analogue of Eq. (3.1). The average velocity is this displacement divided by the elapsed time, $\vec{v}_{av} = \dfrac{\Delta \vec{r}}{t_2-t_1}$, with x and y components $(v_{av})_x = \dfrac{x_2-x_1}{t_2-t_1}$ and $(v_{av})_y = \dfrac{y_2-y_1}{t_2-t_1}$. Average velocity is a vector pointing along the direction of the displacement $\Delta \vec{r}$; in terms of its components, its magnitude and the angle $\theta$ it makes with the x-axis are given by $v_{av} = \sqrt{(v_{av})_x^2 + (v_{av})_y^2}$ and $\tan\theta = \dfrac{(v_{av})_y}{(v_{av})_x}$. …