**Scalars and Vectors**

Learn the difference when describing and using scalars and vectors. See how to use the scalars distance and speed and vectors displacement and velocity mathematically.

**Basic Velocity and Speed Variables**

Name |
Variable |
Unit |
Unit Abbreviation |

Velocity or Speed |
v | meters per second | m/s |

Time |
t | seconds | s |

Displacement or Distance |
X | meters | m |

**How are scalars different than vectors?**

A **scalar** in physics includes only a magnitude. A **magnitude** is a number and a unit. Two common scalars are **distance**, for example 5 meters, and **speed**, for example 5 meters per second. Speed is a scalar because it uses the scalar distance traveled per time.

A **vector** in physics includes a magnitude and a direction. Two common vectors are **displacement** like 5 meters east and **velocity** like 5 meters per second east.

**How is distance different than displacement?**

**Distance** (a scalar) is a measure between two points. Because distance has no direction you add up all of the segments traveled.

**Displacement** (a vector) is a measure of where you are from the origin. So how far in what direction from the **origin** or starting point.

**Scalar Distance and Vector Displacement**

**How do you calculate distance and displacement?**

Look at the difference between **how distance and displacement are calculated**. The stickman in the picture goes right from the origin (0 meters) all the way to 6 meters. Then he turns left and travels 4 meters back. Because direction does not matter, for **distance you add every meter traveled in any direction up.** So 6 meters + 4 meters.

**Distance = 6 + 4 = 10 meters**

For the **displacement vector** you **turn direction into a mathematical sign** and add vectors. Observe the picture to the right showing common signs to replace directions. So we call right positive and left negative. Therefore 6 meters right becomes +6 meters and 4 meters left becomes -4 meters. Now add the vectors but include the signs (+6) + (-4) = +2.

**Displacement**** = (+6) + (-4) = +2. **

Once you have your final answer turn the sign back into a direction and then include the unit. So +2 becomes 2 meters right.

**Calculating Scalar Speed and Vector Velocity**

Scalars go with scalars, so use **distance to calculate speed** as seen in the equation on the right.

Vectors go with vectors, so use **displacement to calculate velocity** as seen in the equation to the right.

The equation in the animation to right (v =Δx/t and also seen in two forms in the picture above it) will be used in examples to calculate speed or velocity but I will use distance if calculating speed and displacement if calculating velocity.

- In different equations x or Δx may be used interchangeably to represent displacement or position.
- Δx would represent change in position (x
_{f}– x_{i}) and displacement is often just written as x in common equations. - A time interval (t) can be calculated (t
_{f}– t_{i})

Notice in the picture to the right the stickman stops and goes at different rates.

**How is instantaneous velocity different than average velocity?**

**Instantaneous velocity**is the velocity at that instant. For example, if stopped, his instantaneous velocity is 0 m/s but a few seconds later he is going again and its 4 m/s.- Total displacement divided by the total time is
**average velocity**. In the animation to the right, the total displacement was +14 m and the time it took was 4 seconds. v = +14/4 becomes 3.5 m/s forward.

**What does acceleration describe?**

**Acceleration** (a **vector**) describes a velocity changes over time. Therefore, the unit for acceleration has the unit for velocity over another time unit. So a meters per second change per second states as **m/s/s** or more commonly **m/s ^{2}**.

You'll learn various means to calculate acceleration in the next section.

Observe the animations below to see how distance and displacement are calculated.

**Scalar Vector Example 1**

Use the picture to the right for the following questions.

**Thomas, on his lunch break, took 30 minutes to go to the library and then the cafe.**

1. What **distance** did Thomas travel during lunch?

2. What was his **displacement**?

3. Determine the **speed** of Thomas during the 30 minute (1800 second) lunch?

4. What was the **velocity** of Thomas during the 30 minutes?

**Links**

- Continue to part
**2: Accelerated Motion and Equations** - Back to the
**Stickman Physics Home Page** - For video tutorials and other physics resources check out
**HoldensClass.com** - Find many of your animation resources in one place at the
**StickMan Physics Gallery** **Equation Sheet**