Speed as the Magnitude of a Velocity Vector

Speed and Velocity Are Not the Same Thing

When a particle moves through the plane, its position can be described by two coordinates that change with time. A convenient way to write its position is with a vector-valued function.

The velocity vector tells you both how fast the particle is moving and the direction in which it is moving. Speed, however, is only the numerical size of that velocity. In other words, speed is the magnitude of the velocity vector.

This distinction is important on the AP Calculus BC exam. A velocity vector can have positive and negative components, but speed can never be negative.

Finding the Velocity Vector

If the position of a particle is

then differentiate each component with respect to time:

The velocity vector is therefore made up of the horizontal and vertical components of velocity. The signs of those components tell you the direction of motion.

Speed Is the Magnitude of Velocity

If the velocity vector is

then its magnitude is found using the Pythagorean theorem:

Therefore, if a particle has horizontal velocity and vertical velocity , its speed is:

Notice that the components are squared. This means that a negative velocity component does not make the speed negative. A particle moving left or downward can still have a positive speed.

The Step-by-Step Method

1. Find the velocity vector. Differentiate the position function component-by-component.
2. Identify the horizontal and vertical velocity components. These are the two entries of the velocity vector.
3. Take the magnitude. Square each component, add them, and take the positive square root.
4. Evaluate at the requested time, if the problem asks for the speed at a particular instant.

Worked Examples

Why the Magnitude Formula Works

Think of the velocity vector as a right triangle. Its horizontal component gives one leg of the triangle, and its vertical component gives the other leg. The velocity vector itself is the hypotenuse.

By the Pythagorean theorem, the length of the velocity vector is the square root of the sum of the squares of its components. That length is exactly the particle's speed.

This is why speed is always nonnegative, even when one or both components of velocity are negative.

Speed vs. Velocity

The difference is worth memorizing:

• Velocity is a vector. It includes direction and can have positive or negative components.
• Speed is a scalar. It measures only how fast the particle is moving and is always nonnegative.
• Speed = magnitude of velocity.

Common Mistakes

Practice Problems