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.
Understanding Calculus: The Net Change and Distance
A velocity graph has a useful built-in scale. If time is measured in seconds and velocity is measured in metres per second, each small rectangle between the graph and the time axis represents a number of metres. Its width is a short time interval.
Its height is a representative velocity during that interval. Multiplying those values gives a small change in position. Adding many rectangles gives an approximation.
Making the rectangles narrower improves it. The integral is the exact value reached in the limit of this process. This explains why the result has units of length, not units of velocity or time.
Position depends on where motion began. A result from the velocity graph tells how far the final position is from the starting position, with direction included. For example, a student can walk thirty metres east, then thirty metres west.
The final position matches the starting position, even though the student walked sixty metres. This distinction appears in everyday tools. An odometer records how much road a car has covered.
A map app comparing two locations is concerned with the change in location. Returning home can produce no overall change in position while still involving a long journey.
Turning points need careful attention. A velocity of zero means the object is momentarily at rest. It does not always mean the object reverses direction.
A graph may touch the time axis and stay on the same side, like a car that briefly stops before continuing forward. A true reversal happens when the graph crosses the axis. When finding total distance, each section of motion must contribute a positive amount.
On a graph made from straight segments, the needed regions are often rectangles, triangles, or trapezoids. For a curved graph, students may use an antiderivative or a numerical estimate. Finding every time where velocity is zero prevents parts of the journey from being counted with the wrong sign.
Average velocity can be surprising because opposite motions can cancel. A trip with a large amount of forward motion followed by nearly equal backward motion may have a small average velocity. Average speed answers a different practical idea.
It uses the full path length divided by total elapsed time, so it cannot be negative. Do not confuse negative velocity with slowing down. Negative velocity describes direction relative to a chosen positive direction.
Whether an object speeds up or slows down depends on velocity together with acceleration. When reading any motion graph, check the axes, units, time interval, and places where the graph reaches or crosses zero.
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.