Total Distance Traveled by a Particle in the Plane

Distance Is Not the Same as Displacement

When a particle moves in the plane, its position can be described by a vector-valued function , where gives the particle's horizontal position and gives its vertical position at time .

The displacement of the particle over an interval is the change in its position. It tells you where the particle ended up relative to where it started. But displacement does not tell you how much ground the particle actually covered.

If a particle moves forward, turns around, and then comes back toward its starting point, its displacement may be small even though it traveled a substantial distance.

Velocity Gives Us the Direction and Speed

The velocity vector is the derivative of the position vector:

Therefore, if , then:

The velocity vector tells us both how the particle is moving horizontally and how it is moving vertically. But for total distance, we need the particle's speed.

Speed Is the Magnitude of the Velocity Vector

If the velocity vector is , its magnitude is:

This is the particle's speed. Speed is always nonnegative because it represents how quickly the particle is moving, regardless of its direction.

Notice the squares under the square root. The horizontal and vertical velocity components contribute together to the particle's instantaneous speed.

The Total Distance Formula

Once we know the speed as a function of time, we can accumulate that speed over the entire time interval. If the particle moves from time to time , the total distance traveled is:

In other words:

This is one of the most important formulas for vector-valued motion in AP Calculus BC.

Why We Integrate Speed

Think about what happens during a very small time interval . If the particle's speed is approximately , then the small amount of distance traveled is approximately:

Adding all of those tiny distances over the entire interval produces a definite integral:

So the distance formula is really just the continuous version of the familiar relationship distance = rate × time.

A Step-by-Step Method

1. Start with the position function. Identify and .
2. Differentiate to find velocity. Calculate and .
3. Find speed. Take the magnitude of the velocity vector: .
4. Set up the distance integral. Integrate speed over the requested time interval.
5. Evaluate the integral. Use an antiderivative when possible or a calculator when the integral does not have a convenient elementary antiderivative.

Worked Examples

The Most Important AP Calculus Distinction

Students frequently confuse net displacement with total distance traveled. The formulas look similar, but there is a critical difference.

The displacement integral uses the velocity vector itself. The distance integral uses the magnitude of the velocity vector.

In one-dimensional motion, this distinction appears as the difference between integrating velocity and integrating the absolute value of velocity:

In two-dimensional motion, the same idea applies, except the absolute value is replaced by the magnitude of the velocity vector.

Why You Cannot Usually Just Integrate the Velocity Components

Suppose . It is tempting to calculate and add the two integrals.

That does not give total distance. The two components describe movement in perpendicular directions, so their contributions must be combined using the Pythagorean theorem at each instant.

Only after finding this magnitude do we integrate over time.

When the Particle Changes Direction

In two-dimensional motion, a particle can change direction even when neither velocity component individually becomes zero. The direction of the velocity vector is determined by both components together.

For total distance, this does not create the same absolute-value issue that occurs in one-dimensional motion. The speed is already a magnitude:

Because a square root of a sum of squares is nonnegative, the speed automatically measures the amount of motion regardless of direction.

Using a Calculator for Distance

Many AP Calculus BC distance problems lead to an integral that does not have a convenient elementary antiderivative. That is completely normal.

For example, if the speed is , you can set up:

Then use your calculator's numerical integration function to evaluate the integral. On the AP exam, the setup is often just as important as the numerical value.

A Reliable AP Calculus Workflow

1. Write the position vector.
2. Differentiate to get the velocity vector.
3. Take the magnitude of velocity.
4. Put the speed inside a definite integral.
5. Use the stated time interval as the limits.
6. Evaluate exactly or numerically.
7. Include units when the problem gives them.

A useful memory shortcut is:

Practice Problems