2.1.1.3. Speed and Velocity

Table of Contents

Understanding Motion Rate

When an object moves, we often want to know how fast it moves and in which direction it moves. Two closely related ideas help us describe this, speed and velocity. They are not the same, even though in everyday language people often use them as if they were.

Speed tells us how fast something is moving. Velocity tells us how fast it is moving and also the direction of motion along the line.

In one dimensional motion, the direction is usually described with a sign. For example, motion to the right may be taken as positive, and motion to the left as negative. Then velocity can be positive or negative, while speed is never negative.

Speed

Speed is based on the distance traveled in a certain amount of time. If a car travels a total distance of 100 meters in 5 seconds, its speed describes how quickly that distance was covered.

The basic idea is

$$\text{speed} = \frac{\text{distance}}{\text{time}}$$

Since distance is always zero or positive, speed is also always zero or positive.

A stationary object has speed zero. A moving object has positive speed.

Important idea:
Speed depends on total distance traveled, not on direction.
$$\text{speed} = \frac{\text{distance}}{\text{time}}$$
Speed can never be negative.

Velocity

Velocity is based on displacement, not distance. Displacement includes direction, so velocity also includes direction.

In one dimension,

$$\text{velocity} = \frac{\text{displacement}}{\text{time}}$$

If an object moves in the positive direction, its velocity is positive. If it moves in the negative direction, its velocity is negative.

For example, if a person walks 20 meters to the right in 4 seconds, the velocity is

$$v = \frac{20\ \text{m}}{4\ \text{s}} = 5\ \text{m/s}$$

If instead the person walks 20 meters to the left, and left is chosen as negative, then the displacement is $-20\ \text{m}$ and

$$v = \frac{-20\ \text{m}}{4\ \text{s}} = -5\ \text{m/s}$$

The negative sign does not mean slow. It means the motion is in the negative direction.

Important idea:
Velocity depends on displacement, so it includes direction.
$$\text{velocity} = \frac{\text{displacement}}{\text{time}}$$
In one dimensional motion, the sign of velocity shows direction.

Speed and Velocity Compared

Speed and velocity can have the same numerical value, but they do not mean the same thing. The key difference is direction.

Quantity	Based on	Includes direction	Can be negative
Speed	Distance	No	No
Velocity	Displacement	Yes	Yes

This difference becomes very important when an object changes direction.

Suppose a student walks 3 meters to the right, then 3 meters back to the left. The total distance traveled is 6 meters, but the final displacement is 0 meters because the student ends where they started.

Over the whole trip, the speed is based on 6 meters of travel, while the velocity is based on 0 meters of displacement.

So an object can have nonzero speed but zero velocity over a time interval.

Units

The SI unit for both speed and velocity is meters per second, written as

$$\text{m/s}$$

Other common units are kilometers per hour, written as $\text{km/h}$, and centimeters per second, written as $\text{cm/s}$. No matter which unit is used, the meaning of speed and velocity remains the same.

Quantity	Common SI unit
Speed	$\text{m/s}$
Velocity	$\text{m/s}$

Direction in One Dimension

In one dimensional motion, direction is usually represented in one of two ways. We can use words such as right and left, or up and down. We can also use plus and minus signs.

For example, if right is positive, then

$$v = +4\ \text{m/s}$$

means motion to the right, and

$$v = -4\ \text{m/s}$$

means motion to the left.

The choice of positive direction is arbitrary, but once chosen, it must be used consistently.

Rule:
Choose a positive direction first. Then interpret the sign of velocity using that choice.
A positive velocity means motion in the positive direction.
A negative velocity means motion in the negative direction.

Everyday Examples

A car moving along a straight road at 15 m/s east has a speed of 15 m/s and a velocity of 15 m/s east.

A runner going around a track may keep a high speed, but because the direction changes, the velocity changes too. In one dimensional motion, this can be seen when someone moves forward and then backward.

An elevator going upward at constant speed has a positive velocity if upward is chosen as positive. The same elevator going downward with the same speed has a negative velocity.

Simple Numerical Example

Imagine a toy car moves 8 meters to the right in 2 seconds.

Its speed is

$$\text{speed} = \frac{8\ \text{m}}{2\ \text{s}} = 4\ \text{m/s}$$

Its velocity is also

$$\text{velocity} = \frac{8\ \text{m}}{2\ \text{s}} = +4\ \text{m/s}$$

Now imagine the toy car moves 8 meters to the left in 2 seconds, with right chosen as positive.

Its speed is still

$$4\ \text{m/s}$$

because speed ignores direction.

But its velocity is

$$\text{velocity} = \frac{-8\ \text{m}}{2\ \text{s}} = -4\ \text{m/s}$$

Visualizing Positive and Negative Velocity

Rendering scientific sketch…

Motion in one dimension and sign of velocity

Rest and Motion

If an object is not changing its position, its speed is zero and its velocity is zero.

If the object is moving, its speed is greater than zero. Its velocity may be positive or negative depending on the direction of motion.

This makes velocity especially useful in physics, because it tells us more than speed alone.

Key Distinction

The most important thing to remember is that speed is a scalar quantity, while velocity in one dimension acts like a signed quantity because it carries directional information. Later, this directional nature becomes even more important in higher dimensions.

Most important distinction:
Speed tells how fast.
Velocity tells how fast and in what direction.
For motion along one straight line,
$$\text{speed} = \frac{\text{distance}}{\text{time}}, \qquad
\text{velocity} = \frac{\text{displacement}}{\text{time}}$$

Final Intuition

If you only care about how rapidly an object moves, use speed. If you care about both motion and direction, use velocity. In physics, velocity is often the more informative quantity, because motion is not just about how much ground is covered, but also about where the object is going.

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