
Next time you are in a car, watch the speedometer and think about what it tells you. Is the reading on the speedometer different if you turn around and drive in the opposite direction at the same speed? What does the speedometer read when the car is backing up? A speedometer reads a scalar value—a speed. To describe forward and backward motion we need a vector—the velocity vector.

Speed and velocity

A complete threedimensional (3D) description of a car trip requires three coordinates: x, y, and z.
The car’s position along a road, however, can be described more simply by using a single value—such as mile post 167. When position can be described by a single value we say the description is one dimensional.


A onedimensional (1D) definition of speed is the ratio of distance traveled to time taken.
If you go 120 miles in 2 hours your speed is the distance divided by the time, or 120 ÷ 2 = 60 miles per hour. In physics, we will typically use meters and seconds instead of miles and hours. Since 60 miles is 96,558 meters, and 1 hour = 3,600 seconds, 60 mph is the same speed as 26.8 meters per second (m/s).

(3.2)  $$v=\frac{d}{t}$$
 v  =  speed (m/s)  d  =  distance traveled (m)  t  =  time taken (s) 
 Speed


An object moving with a speed of 1 m/s changes its position by one meter each second. This is about the speed of your hand if you sweep it across a table in the time it takes you to say “one thousand.” A car going 65 mph on the highway is moving with a speed of 29 m/s. That is roughly equal to moving three times the length of an average classroom in one second. The speeds we will consider in physics range from zero when an object is standing still to 300,000,000 m/s, which is the speed of light.

Equation (3.2) is limited because a distance can be zero or positive but not negative, and therefore speeds are always positive. To account for moving backward we extend the concept of speed to include velocity v. Velocity is defined by equation (3.3):

(3.3)  $$\overrightarrow{v}=\frac{\text{\Delta}x}{\text{\Delta}t}\text{\hspace{1em}}\text{or}\text{\hspace{1em}}\overrightarrow{v}=\frac{\overrightarrow{d}}{\text{\Delta}t}$$
 $\stackrel{\rightharpoonup}{v}$  =  velocity (m/s)  Δx  =  displacement (m)  $\stackrel{\rightharpoonup}{d}$  =  displacement (m)  Δt  =  change in time (s) 
 Velocity


The symbol “Δ” translates to “the change in” and is pronounced “delta.” Since x is position, Δx means “the change in position,” which is identical to the displacement d. Equation (3.3) is a better definition because velocity is the change in position divided by the change in time. Because positions can be in front or behind, velocity can be positive or negative depending on direction.
In short, velocity is a vector with both direction and magnitude—which can take on negative values—whereas speed is a scalar that represents only the magnitude of the velocity vector.
