# An object travels North at  4 m/s for 1 s and then travels South at  5 m/s for  7 s. What are the object's average speed and velocity?

Apr 21, 2018

${V}_{a v g} = - 3.875 \frac{m}{s}$

${S}_{a v g} = 4.875 \frac{m}{s}$

#### Explanation:

The average velocity is a vector quantity and therefore the signs of the velocities are taken into consideration.

The first step is to find the two velocities:

Here I will define North as the positive direction and South as the negative direction, however you could define your system however you want.

North at $4 \frac{m}{s}$ is $4 \frac{m}{s}$

South at $5 \frac{m}{s}$ is $- 5 \frac{m}{s}$

From here, you take the average of these two measurements while considering time as well.

${V}_{a v g} = \frac{\text{displacement}}{\Delta t} = \frac{{v}_{1} {t}_{1} + {v}_{2} {t}_{2}}{\Delta t}$

${V}_{a v g} = \frac{4 \frac{m}{s} \cdot 1 s - 5 \frac{m}{s} \cdot 7 s}{8 s} = - 3.875 \frac{m}{s}$

A good way to understand this is to think about how averages are taken. You add up the total amount of all the items, then divide by the number of items. This is the same principle here: we are adding up all the 1 second intervals of velocity then dividing by the total number of 1 second intervals. You can also think of it as measuring the displacement ($v \cdot t$) and dividing by the time it took to get there.

The average speed is a scalar quantity and therefore the signs of the velocities are not taken into consideration.

The most intuitive way to find average speed is to find the total distance traveled. In the case of this problem, where the direction of travel is only up or down, the following equation can be used to find average speed:

${S}_{a v g} = \frac{\text{distance}}{\Delta t} = \frac{\left\mid {v}_{1} {t}_{1} \right\mid + \left\mid {v}_{2} {t}_{2} \right\mid}{\Delta t}$

${S}_{a v g} = \frac{\left\mid 4 \frac{m}{s} \cdot 1 s \right\mid + \left\mid - 5 \frac{m}{s} \cdot 7 s \right\mid}{8 s} = 4.875 \frac{m}{s}$