# Speed and Velocity

## Speed and Velocity

Speed is how fast something moves.

Velocity is speed with a **direction**.

Saying Ariel the Dog runs at **9 km/h** (kilometers per hour) is a speed.

But saying he runs **9 km/h Westwards** is a velocity.

Speed | Velocity | |
---|---|---|

Has: |
magnitude |
magnitude and direction |

Example: | 60 km/h | 60 km/h North |

Example: | 5 m/s | 5 m/s upwards |

Imagine something moving back and forth very fast: it has a high speed, but a low (or zero) velocity.

## Speed

Speed is measured as distance moved over time.

Speed = \frac{Distance}{Time}

### Example: A car travels 50 km in one hour.

Its average speed is 50 km per hour (50 km/h)

Speed = \frac{Distance}{Time} = \frac{50 km}{1 hour}

We can also use these symbols:

Speed = \frac{Δs}{Δt}

Where Δ ("*Delta*") means "change in", and

- s means distance ("s" for "space")
- t means time

### Example: You run 360 m in 60 seconds.

Speed = | \frac{Δs}{Δt} | |

= | \frac{360 m}{60 seconds} | |

= | \frac{6 m}{1 second} |

So your speed is 6 meters per second (6 m/s).

## Units

Speed is commonly measured in:

- meters per second (m/s or m s
^{-1}), or - kilometers per hour (km/h or km h
^{-1})

A km is 1000 m, and there are 3600 seconds in an hour, so we can convert like this (see Unit Conversion Method to learn more):

\frac{1 m}{1 s} × \frac{1 km}{1000 m} × \frac{3600 s}{1 h} = \frac{3600 m · km · s}{1000 s · m · h} = \frac{3.6 km}{1 h}

So **1 m/s is equal to 3.6 km/h**

### Example: What is 20 m/s in km/h ?

20 m/s × \frac{3.6 km/h}{1 m/s} = 72 km/h

### Example: What is 120 km/h in m/s ?

120 km/h × \frac{1 m/s}{3.6 km/h} = 33.333... m/s

## Average vs Instantaneous Speed

The examples so far calculate **average speed**: how far something travels over a period of time.

But speed can change as time goes by. A car can go faster and slower, maybe even stop at lights.

So there is also ** instantaneous speed**: the speed at an **instant** in time. We can try to measure it by using a very short span of time (the shorter the better).

### Example: Sam uses a stopwatch and measures 1.6 seconds as the car travels between two posts 20 m apart. What is the ** instantaneous speed**?

Well, we don't know exactly, as the car may have been speeding up or slowing down during that time, but we can estimate:

\frac{20 m}{1.6 s} = 12.5 m/s = 45 km/h

It is really still an average, but is close to an instantaneous speed.

## Constant Speed

When the speed does not change it is **constant**.

For constant speed, the average and instantaneous speeds are the same.

## Velocity

Velocity is speed with a **direction**.

It is actually a vector ...

... as it has magnitude **and** direction

Because the direction is important velocity uses **displacement** instead of distance:

Speed = \frac{Distance}{Time}

Velocity = \frac{Displacement}{Time} in a direction.

### Example: You walk from home to the shop in 100 seconds, what is your speed and what is your velocity?

Speed = \frac{220 m}{100 s} = 2.2 m/s

Velocity = \frac{130 m}{100 s} East = 1.3 m/s East

### You forgot your money so you immediately return home in 120 more seconds, what is your round-trip speed and velocity?

The total time is 100 s + 120 s = 220 s:

Speed = \frac{440 m}{220 s} = 2.0 m/s

Velocity = \frac{0 m}{220 s} = 0 m/s

Yes, the velocity is zero as you ended up where you started.

Learn more at Vectors.

## Relative

Motion is relative. When we say something is "at rest" or "moving at 4 m/s" we forget to say "in relation to me" or "in relation to the ground", etc.

Think about this: are you really standing still? You are on planet Earth which is spinning at 40,075 km per day (about 1675 km/h or 465 m/s), and moving around the Sun at about 100,000 km/h, which is itself moving through the Galaxy.

Next time you are out walking, imagine you are still and it is the world that moves under your feet. Feels great.

**It is all relative!**