Lesson 5.02 Speed, Velocity, and Acceleration

Speed

To describe how fast an object moves, we use its speed. Speed is the distance an object travels per unit time. We calculate speed by using the following:

Speed equals distance divided by time.

The equation for speed is:

s = d / t

The unit of measurement for speed is m/s, or meters per second. Sometimes speed is expressed:

• km/h or kilometer per hour
• mph or miles per hour (the English Standard)

Speed does not take into account the direction an object is traveling. Therefore, speed is a scalar quantity a quantity with magnitude but no direction .

Since speed is a scalar, it is described as having a magnitude and unit only. Magnitude is the number value. For example, Kyle rode his bike 5 m/s down the street.

• The "5" describes the magnitude.
• The m/s is the unit of measurement for speed.

Solving for Other Variables

This equation can be rewritten to solve for the total time or the total distance when the other two variables are known.

To solve for total time, we can rearrange the equation to read time equals distance divided by speed, or t = d / s

To solve for total distance, we can rearrange the equation to read distance equals speed multiplied by time, or d = s x t

Constant Speed

When an object is travelling at a constant speed, it does not speed up or slow down. The speed is the same regardless of the time interval chosen.

Most of the time, speed is not constant. For example, when you get in your car, you do not travel at the same speed the entire time. You speed up, then you slow down. You may stop and then speed up again. Because speed changes, we often talk about either average speed or instantaneous speed. We will look at both of these quantities in the next two sections of the lesson.

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Average Speed

Average speed is the total distance traveled divided by the total time of travel. Average speed is calculated using the same formula as speed.

We denote average speed with the symbol s_avg, and we solve for average speed using the following:

savg equals total distance (dT) divided by total time (tT)

The formula is written:

savg = dT / tT

To find the total distance, or d_T, you can add all of the distances traveled in a scenario.

To find the total time, or t_T, you can add all of the time traveled.

Once you have these totals, you can divide the total distance by the total time to find the average speed of a particular journey.

Example

Let's try to solve for average speed in the following example. If a car travels 90 km in the first hour of a two-hour trip and then travels 110 km in the second hour, what is the car's average speed for the entire trip?

1. First, let's identify the known values:

   The first piece of information we're given indicates the car traveled a distance of 90 km in the first hour of the trip. From this information, we know:

   • d₁ = 90 km
   • t₁ = 1 h

   The second piece of information we're given says the car traveled 110 km in the second hour of the trip. From this information, we know:

   • d₂ = 110 km
   • t₂ = 1 h

2. Next, we need to combine the two distances traveled to find the total distance.
   • d₁ + d₂= Total distance
   • 90 km + 110 km = 200 km
   • 200 km = Total distance


3. Now we need to combine the two times to find the total time traveled.
   • t₁ + t₂ = Total time
   • 1 h + 1 h = 2 h
   • 2 h = Total time


4. Let's plug the total time and total distance into the average speed formula and solve.
   • s_avg = d_T / t_T
   • s_avg = 200 km / 2 h
   • s_avg = 100 km/h

In this scenario, the average speed of the car was 100 km/h.

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Instantaneous Speed

Instantaneous speed is how fast you are going at one point in time or one instant. A vehicle's speedometer shows instantaneous speed.

• If an object is speeding up or slowing down, its instantaneous speed is different at every point in time.
• When an object is moving with constant speed, the instantaneous speed does not change. The speed is the same at each point in time.

In general, there is no calculation for instantaneous speed since it is simply the speed at one specific point in time. It is reported simply using a magnitude and a unit. For example, the car was traveling at 6 m/s.

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Velocity

Hopefully the concept of speed we just discussed was pretty familiar to you. After all, you have probably ridden in a car or some other type of vehicle most days of your life!

Next, we will move on to the concept of velocity. Velocity is very similar to speed EXCEPT that velocity is a vector quantity, which means it includes direction as well as speed.

Velocity includes an object's:

• Speed
• Unit
• Direction of motion

The only difference between speed and velocity is that velocity requires direction. It is possible for two objects to have the same speed yet different velocities.

For example, in the image above, the two elevators have the same speed yet they are moving in opposite directions. One elevator moves up while the other moves down. The elevator going up is denoted with a positive (+) direction, while the elevator going down is denoted with a negative (-) direction. The elevators have the same speed but different velocities. The one going up has a positive velocity and the one doing down has a negative velocity.

The formula we use to solve velocity is much like that of speed. Do you notice the only difference?

We denote velocity with the symbol v, and we solve for velocity using the following:

Velocity equals distance divided by time.

The formula is written:

v = d / t

The unit of measurement for velocity is m/s, or meters per second. Sometimes velocity is expressed:

• km/h
• mph or miles per hour

Since velocity is a vector, it is described as having:

• Magnitude
• Unit
• Direction of motion

For example, Kyle rode his bike 5 m/s east on Main Street.

• The "5" describes the magnitude.
• The "m/s" is the unit of measurement for velocity.
• The direction provided is east.

As a final note, many times the terms “speed” and “velocity” are used interchangeably, especially in scenarios involving math or numbers. It's easy to see why people do this. Velocity and speed are calculated in basically the exact same way — you divide the distance by the time. However, please remember, in science courses like this one, velocity must be reported with a direction. This direction could be the actual cardinal direction north, south, east, west or with a plus (+) or minus (-) sign.

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Acceleration

The final aspect of motion we will discuss in this lesson is acceleration. Acceleration is another vector quantity a quantity that includes magnitude and direction .

Acceleration is the rate of change of velocity. Acceleration occurs when there is a change in speed or direction of an object or both.

Accelerating objects can be:

• Speeding up (also called positive acceleration)
• Slowing down (also called negative acceleration)
• Changing direction

Many times in our everyday conversations we use the term “acceleration” to indicate a change in speed. Consider the following example statement: “The car accelerated when the light turned green.” This statement is correct; however, as a physical science student, you must also remember that if a car changes directions and maintains the same speed, it is also an example of acceleration.

Acceleration can also mean slowing down. I know that's very different from how we use the term normally. Since acceleration is simply a change in velocity, that change could be to speed up or slow down.

Since acceleration is a vector, it must be written using:

• Magnitude
• Unit
• Direction

Image attribution: "Cycle Racing 1997" by Martin Pettitt is licensed under CC BY 2.0. Image has been changed from original.

Look at the cyclists in this picture along with the arrows above them.

• The image on the left shows that if the velocity and acceleration are in the same direction, the cyclists will be going faster.
• However, if acceleration and velocity are in opposite directions (shown in the image on the right), the cyclists slows down.

Formula for Acceleration

We calculate acceleration by using the following:

Acceleration equals change in velocity divided by time.

The formula is written:

a = Δv / t

In this formula Δv means the change in velocity (the delta symbol Δ means "change in").

You can also calculate acceleration using:

a = (final velocity - initial velocity) / time

or

a = (vf - vi) / t

In this formula:

• v_f = final velocity
• v_i = initial velocity

Let's look at an example problem: Sara covers the last straight stretch of a race in 6 s. During that time, she speeds up from 4 m/s to 10 m/s. What was her acceleration?

1. First let's identify the known values.
   • v_f = 10 m/s
   • v_i = 4 m/s
   • t = 6 s


2. Next, we can plug the known values into our formula.

   a = (v_f – v_i) / t

   a = (10 m/s – 4 m/s) / 6 s

   a = (10 m/s – 4 m/s) / 6 s

   a = 1 m/s²

Therefore, Sara accelerated at 1 m/s².

In terms of units, acceleration (for a change in speed) is always measured in m/s².

To learn more about the relationship between velocity and acceleration, open the video Kinematics (0:14). Login information.

• First, read the Background Essay listed under the Support Materials heading.
• Then, watch the video titled “Kinematics”

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