1.3 Displacement and Velocity

Learning Objectives

After Chapter 1.3, you will be able to:

• Describe the relationship between the average and instantaneous versions of velocity and speed
• Distinguish between total distance and total displacement
• Connect displacement and velocity with an equation
  

Now that we’ve covered the basic geometry that serves as the foundation of physics, we can examine the related physical quantities. The basic quantities that relate to kinematics are displacement, velocity, and acceleration.

Displacement

An object in motion may experience a change in its position in space, known as displacement (x or d). This is a vector quantity and, as such, has both magnitude and direction. The displacement vector connects (in a straight line) the object’s initial position and its final position. Understand that displacement does not account for the actual pathway taken between the initial and the final positions—only the net change in position from initial to final. Distance (d) traveled, on the other hand, considers the pathway taken and is a scalar quantity.

Example:

What is the displacement of a man who walks 2 km east, then 2 km north, then 2 km west, and then 2 km south?

Solution:

While his total distance traveled is 8 km, his displacement is a vector quantity that represents the change in position. In this case, his displacement is zero because the man ends up the same place he started, as shown below.

Velocity

As was mentioned earlier, velocity (v) is a vector. Its magnitude is measured as the rate of change of displacement in a given unit of time, and its SI units are meters per second. The direction of the velocity vector is necessarily the same as the direction of the displacement vector. Speed (v) is the rate of actual distance traveled in a given unit of time.

The distinction is subtle, so let’s examine this a little more carefully. The instantaneous speed of an object will always be equal to the magnitude of the object’s instantaneous velocity, which is a measure of the average velocity as the change in time (Δt) approaches zero:

Equation 1.6

where v is the instantaneous velocity, Δx is the change in position, and Δt is the change in time. As a measure of speed, instantaneous speed is a scalar number. Average speed will not necessarily always be equal to the magnitude of the average velocity. This is because average velocity is the ratio of the displacement vector over the change in time (and is a vector), whereas average speed (which is scalar) is the ratio of the total distance traveled over the change in time. Average speed accounts for actual distance traveled, whereas average velocity does not:

Equation 1.7

where is the average velocity, Δx is the change in position, and Δt is the change in time.

Consider the example given earlier regarding the Earth’s orbit. In one year, the Earth travels roughly 940 million kilometers, but its displacement is zero:

The average speed is a measure of distance traveled in a given period of time; the average velocity is a measure of the displacement of an object over a given period of time. While the average speed of the Earth over a year is about 30 kilometers per second, its average velocity is again zero:

MCAT Concept Check 1.3

Before you move on, assess your understanding of the material with these questions.

1. What is the relationship between instantaneous velocity and instantaneous speed? Between average velocity and average speed?

2. True or False: Total distance traveled can never be less than the total displacement.

3. Provide a definition for displacement or velocity in terms of the other variable.