Position, Velocity, and Acceleration of a Particle in the Plane

The Big Picture

In one-dimensional motion, you can describe a particle's location with a single function such as . In the plane, a particle needs two coordinates: an x-coordinate and a y-coordinate.

The natural way to represent both coordinates at once is with a vector-valued position function. Once you know the position function, differentiation gives you velocity, and differentiating velocity gives you acceleration.

This creates a simple chain of ideas:

The most important skill is keeping track of what each vector represents and what its components mean.

Position in the Plane

If a particle moves in the plane, its position at time can be written as

The function gives the horizontal coordinate, while gives the vertical coordinate.

For example, if

then at time , the particle is at the point .

Velocity: The Rate of Change of Position

Velocity tells you both how fast the particle is moving and which direction it is moving. It is the derivative of the position vector.

In component form,

So the x-component of velocity is the rate of change of the x-coordinate, and the y-component is the rate of change of the y-coordinate.

Speed Is the Magnitude of Velocity

A common AP Calculus trap is confusing velocity with speed. Velocity is a vector; speed is a scalar.

If the velocity is , then the speed is obtained using the Pythagorean theorem.

Speed is therefore always nonnegative, while a velocity component can be positive, negative, or zero.

Acceleration: The Rate of Change of Velocity

Acceleration measures how velocity changes over time. It is the derivative of the velocity vector and the second derivative of position.

In component form,

Notice that acceleration does not simply mean "the particle is speeding up." A particle can accelerate while its speed decreases if the acceleration points opposite the direction of motion.

Worked Examples

Example 1: Find Position, Velocity, and Acceleration

A particle moves in the plane with position function

Find the position, velocity, and acceleration at .

Step 1: Position
Substitute into the position function.

Step 2: Velocity
Differentiate each component of the position vector.

Step 3: Evaluate velocity

Step 4: Acceleration
Differentiate the velocity vector.

Step 5: Evaluate acceleration

Result
At the specified time, the particle's position is , its velocity is , and its acceleration is .

Example 2: Find the Speed of a Particle

A particle has velocity

Find the speed of the particle at .

Step 1: Evaluate the velocity components

Step 2: Take the magnitude

Result
The particle's speed is .

Example 3: Determine Whether the Particle Is Speeding Up

Suppose the velocity and acceleration of a particle at a particular instant are

Is the particle speeding up or slowing down?

Step 1: Compare velocity and acceleration
The dot product is

Step 2: Interpret the sign
A positive dot product means that acceleration has a component in the same direction as velocity.

Result
Because is positive, the particle is speeding up.

Example 4: Motion in the Plane from Component Functions

A particle's coordinates are given by

Find the velocity, acceleration, and speed at .

Step 1: Differentiate position

Step 2: Differentiate again

Step 3: Evaluate velocity

Step 4: Evaluate acceleration

Step 5: Find speed

Result
At , the particle has velocity , acceleration , and speed .

How to Interpret the Components

The component form is especially useful on AP Calculus free-response questions.

Here is what each quantity tells you:

: horizontal position.
: vertical position.
: horizontal velocity.
: vertical velocity.
: horizontal acceleration.
: vertical acceleration.

For example, if , the particle is moving to the right at that instant. If , it is moving downward.

Speeding Up vs. Slowing Down

One of the more subtle AP Calculus BC ideas is determining whether a particle is speeding up or slowing down.

Compare the directions of velocity and acceleration. If they point generally in the same direction, the particle speeds up. If they point generally in opposite directions, the particle slows down.

A convenient algebraic test uses the dot product:

If , the particle is speeding up.
If , the particle is slowing down.
If , the instantaneous rate of change of speed is zero.

Common Mistakes

Confusing velocity with speed

Velocity is a vector and includes direction. Speed is the magnitude of velocity. If the problem asks for speed, calculate , not just one component of velocity.

Forgetting that position has two components

For motion in the plane, position is a vector. You must differentiate both the x- and y-components when finding velocity and acceleration.

Using position when the problem asks for velocity

Position tells you where the particle is. Velocity tells you how its position is changing. Always check which derivative the question is asking for.

Assuming positive acceleration means speeding up

Acceleration is a vector. A positive x- or y-component by itself does not tell you whether the particle is speeding up. Compare velocity and acceleration, or use their dot product.

Practice Problems

Question 1

A particle has position function . Find its velocity function.

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Question 2

A particle has position function . Find its acceleration at .

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Question 3

A particle has velocity . Find its speed at .

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Question 4

At a particular time, a particle has velocity and acceleration . Is the particle speeding up or slowing down?

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