Calculating Average Velocity From Position Function Graph

What is Average Velocity?

Average velocity is defined as the displacement of an object divided by the time interval during which that displacement occurs. Unlike speed, which is always positive, velocity can be negative if the object moves in the opposite direction of the positive axis.

In mathematical terms, average velocity is a vector quantity that provides information about both the speed and direction of motion. It's particularly useful when analyzing motion over a specific time period, as it gives a single value that represents the overall motion.

Calculating from a Position Function Graph

When you have a position function graph, calculating average velocity involves these steps:

1. Identify two points on the graph that represent the initial and final positions
2. Determine the corresponding times for these positions
3. Calculate the change in position (displacement)
4. Calculate the change in time
5. Divide the displacement by the time interval to get average velocity

The key insight is that the slope of the position-time graph between two points represents the average velocity during that time interval.

The Formula

This formula works for both linear and non-linear position functions, as long as you can identify the specific points on the graph.

Worked Example

Let's say you have a position function graph where:

• At time \( t_1 = 2 \) seconds, the position is \( x_1 = 5 \) meters
• At time \( t_2 = 6 \) seconds, the position is \( x_2 = 17 \) meters

To calculate the average velocity between these two points:

1. Calculate the change in position: \( \Delta x = 17 - 5 = 12 \) meters
2. Calculate the change in time: \( \Delta t = 6 - 2 = 4 \) seconds
3. Divide displacement by time: \( \bar{v} = \frac{12}{4} = 3 \) m/s

The average velocity during this interval is 3 meters per second.

Interpreting the Result

When you calculate average velocity from a position function graph, the result tells you:

• The overall direction of motion (positive or negative value)
• The average speed during the time interval (absolute value of velocity)
• Whether the object was speeding up or slowing down during the interval

For example, if you get a negative average velocity, it means the object was moving in the opposite direction of the positive axis on average during that time period.