# AVERAGE VELOCITY FORMULA

The average velocity is defined as the displacement divided by the time elapsed. Average velocity formula is:

$$average{ \quad }velocity=\frac { displacement }{ time{ \quad }elapsed } =\frac { final\quad position\quad -\quad initial\quad position }{ time\quad elapsed }$$

Velocity is a vector quantity and include not only the magnitude of how fast an object is moving but also the direction in which it is moving.

Figure 1 shows one dimensional motion of an object along a straight line over a particular time interval. It shows the positions of the object (xt) where it is located at the moment in time (t). The positive direction of motion is along the red arrow direction.

The average velocity between time t1 and time t2 is the displacement (xt2-xt1) divided by the time interval (t2-t1) and it’s larger than 0. Average velocity formula in symbolic representation is:

$${\overline {v}}_{{t}_{1},{t}_{2}}=\frac {{x}_{{t}_{2}}-{x}_{{t}_{1}}}{{t}_{2}-{t}_{1}}>0$$.

Between t1 and t5, the average velocity is 0 because x1 and x5 are at the same position and the displacement is 0. Between t2 and t4 , average velocity is smaller than 0 because xt2 minus xt4 is negative.

$${\overline { v } }_{ { t }_{ 1 },{ t }_{ 5 } }=0$$,   $${\overline { v } }_{ { t }_{ 2 },{ t }_{ 4 } }<0$$

The bar (‾) symbol shown over the velocity v is a standard notation for “average”.

Notice that the location of origin 0 on x axis is completely unimportant for the calculation of average velocity. However if the positive direction of motion is chosen in opposite direction, then the signs of average velocity will flip. The direction of motion determines the signs.

Position vs. time diagram in Figure 2 shows positions of the same object not only at discrete moments of time but at any moment of time.

A straight line is drawn between t2 and t3 on the diagram. The slope of the straight line is the ratio ∆x/∆t but this ratio is also the average velocity of the object between t2 and t3. Therefore, the average velocity of an object is equal to the slope of the straight line drawn between final position and initial position during the time interval on an x vs. t plot.

$${ \overline { v } }={ \Delta x }/{ \Delta t }=the\quad slope\quad of\quad the\quad straight\quad line$$

Delta (letter) is used to denote change of any quantity. Then ∆x means “the change in position (x)” and this is the displacement. ∆t means “the change in time (t)” and this is the time elapsed.

If the slope of the line is positive, then the average velocity is positive. If the slope is negative, the average velocity is negative.  For instance, average velocity between tand t3 is positive and between t4 and t5 it’s negative.

If the origin 0 point is changed in the diagram and a different point is selected as 0, we will have found same values for the average velocities. The only difference will be the position of the curve at the plot.