Analyzing Motion: Total Distance vs. Displacement

This lesson introduces how to use definite integrals to analyze motion along a line. You will learn how to compute net displacement and total distance traveled from a velocity function.

Summary

The video explains the difference between velocity and speed, how direction affects displacement, and why absolute value is required when calculating total distance. It shows how to determine when velocity is positive or negative, how to break integrals across sign changes, and how to evaluate both analytically and using Desmos. A second example demonstrates how to compute a new position using an initial condition and an integral of velocity. Students see how to interpret results, include units, and apply correct AP exam notation.

Key Takeaways

• Net displacement equals the integral of velocity over an interval.

• Total distance traveled equals the integral of the absolute value of velocity.

• Velocity can be positive or negative; speed is always non‑negative.

• When velocity changes sign, break the integral at the zeros of the velocity function.

• Negative portions of velocity must be converted to positive when computing total distance.

• Desmos can evaluate definite integrals, absolute value integrals, and show velocity sign changes.

• Knowing initial position plus net displacement gives the new position of an object.

• Correct mathematical notation and units are required to earn credit on the AP exam.