How to Calculate Average Velocity on Intervals

What is Average Velocity?

Average velocity is defined as the displacement of an object divided by the total time taken for that displacement. It's a vector quantity, meaning it has both magnitude and direction. This makes it different from average speed, which only considers the total distance traveled without regard to direction.

In physics, velocity is often represented as a function of time, v(t), and can change at any instant. When calculating average velocity over an interval, we're essentially finding the slope of the position-time graph between two points.

The Formula

The basic formula for average velocity over a time interval is:

For motion with constant velocity, this is straightforward. However, for motion with changing velocity, you'll need to calculate the average over specific time intervals.

How to Calculate Average Velocity on Intervals

Step 1: Identify the Time Intervals

First, determine the specific time intervals for which you want to calculate the average velocity. These could be equal intervals or based on specific events in the motion.

Step 2: Determine Positions at Interval Endpoints

For each interval, find the position of the object at the start and end of the interval. This might come from experimental data, simulations, or known motion equations.

Step 3: Calculate Displacement for Each Interval

Subtract the initial position from the final position for each interval to find the displacement (Δx).

Step 4: Calculate Time Duration for Each Interval

Subtract the initial time from the final time for each interval to find Δt.

Step 5: Compute Average Velocity for Each Interval

Divide the displacement by the time duration for each interval using the formula v_avg = Δx / Δt.

Step 6: Analyze the Results

Examine how the average velocity changes over different intervals. This can reveal patterns in the motion, such as acceleration or deceleration.

Worked Example

Let's calculate the average velocity for a car's motion over two different time intervals.

Scenario

A car moves along a straight road. Its position is recorded at specific times:

• At t = 0 s, x = 0 m
• At t = 2 s, x = 10 m
• At t = 5 s, x = 25 m
• At t = 8 s, x = 40 m

First Interval (0-5 seconds)

Δx = 25 m - 0 m = 25 m

Δt = 5 s - 0 s = 5 s

v_avg = 25 m / 5 s = 5 m/s

Second Interval (5-8 seconds)

Δx = 40 m - 25 m = 15 m

Δt = 8 s - 5 s = 3 s

v_avg = 15 m / 3 s = 5 m/s

In this example, the average velocity is the same for both intervals, but this isn't always the case. The actual average velocity over the entire 8-second period would be (40 m - 0 m)/(8 s - 0 s) = 5 m/s.