When an object moves along a line, its velocity tells both how fast it moves and which direction it moves. Calculus lets us turn a velocity-time graph into a statement about position by adding up tiny changes over time. The signed area under a velocity curve gives net displacement, which is the overall change in position.
This idea matters in physics, engineering, and any situation where motion changes continuously.
Key Facts
- Net displacement from velocity: Δx = ∫ from a to b v(t) dt
- Total distance traveled: distance = ∫ from a to b |v(t)| dt
- Area above the time axis counts as positive displacement.
- Area below the time axis counts as negative displacement.
- If velocity changes sign, split the integral at each zero of v(t).
- Average velocity on [a, b]: v_avg = (1/(b - a)) ∫ from a to b v(t) dt
Vocabulary
- Velocity
- Velocity is the rate of change of position with direction included.
- Displacement
- Displacement is the signed change in position from the starting point to the ending point.
- Total distance
- Total distance is the full length of the path traveled, regardless of direction.
- Definite integral
- A definite integral gives the accumulated signed area under a curve over an interval.
- Absolute value
- Absolute value gives the nonnegative size of a quantity, so |v(t)| represents speed.
Common Mistakes to Avoid
- Using ∫ v(t) dt for total distance, because negative velocity subtracts area and can cancel positive motion.
- Ignoring where v(t) crosses zero, because total distance requires splitting the interval wherever the velocity changes sign.
- Treating negative velocity as negative speed, because speed is always nonnegative and equals |v(t)|.
- Forgetting units, because integrating velocity in meters per second over seconds gives meters, not meters per second.
Practice Questions
- 1 A particle has velocity v(t) = 3t - 6 meters per second from t = 0 to t = 5 seconds. Find its net displacement.
- 2 A particle has velocity v(t) = 4 - t meters per second from t = 0 to t = 6 seconds. Find the total distance traveled.
- 3 A velocity-time graph has equal positive area above the axis and negative area below the axis over a time interval. Explain what this means for net displacement and total distance.