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HELP: Original Lessons on Motion

Site: Mountain Heights Academy OER
Course: Physics Q4
Book: HELP: Original Lessons on Motion
Printed by: Guest user
Date: Friday, 4 April 2025, 11:32 AM

Description

These are the original lessons on motion from Q2.  Review as needed.

Table of contents

Distance vs. Displacement


 

In stockcar races, the winners frequently travel a distance of 500 miles but at the end of the race, their displacement is only a few feet from where they began.

Position, Distance, and Displacement

In order to study how something moves, we must know where it is.  For straight line motion, it is easy to visualize the object on a number line.  The object may be placed at any point on the number line either in the positive numbers or the negative numbers.  It is common to choose the original position of the object to be on the zero mark.  In making the zero mark the reference point, you have chosen a frame of reference.  The position of an object is the separation between the object and the reference point.

When an object moves, we often refer to the amount it moves as the distance.  Distance does not need a reference point and does not need a direction.  If an automobile moves 50 kilometers, the distance traveled is 50 kilometers regardless of the starting point or the direction of movement.  If we wish to find the final position of the automobile, however, just having the distance traveled will not allow us to determine the final position.  In order to find the final position of the object, we need to know the starting point and the direction of the motion.  The change in the position of the object is called its displacement.  The displacement must include a direction because the final position may be either in the positive or negative direction along the number line from the initial position.  The displacement is a vector quantity and vectors are discussed in another section.

Visit the following link for a description of the difference between distance and displacement:

http://www.tutorvista.com/content/physics/physics-i/motion/distance-and-displacement.php

Example 

Problem: An indecisive car goes 120 m North, then 30 m south then 60m North. What is the car's distance and displacement?

Solution:

Distance is the total amount traveled. Thus distance = 120 + 30 + 60 m = 210 m

Displacement is the amount displaced from the starting position. Thus displacement = 120 - 30 + 60 m = 150 m.

Graphing Position and Displacement

Distance vs. Displacement in More Than 1 Dimension

Summary

  • The length of the actual path traveled by an object is called distance.
  • The shortest path between original and final position is called displacement.
  • Distance is a scalar and displacement is a vector.
  • For motion in a straight line, we can simply add/subtract to find distance and displacement.  For motion in multiple dimensions, we can use Pythagorean Theorem and trigonometry to find displacement.

Speed


Did you ever play fast-pitch softball? If you did, then you probably have some idea of how fast the pitcher throws the ball. For a female athlete like the one in the opening image, the ball may reach a speed of 120 km/h (about 75 mi/h). For a male athlete, the ball may travel even faster. A fast-pitch pitcher uses a “windmill” motion to throw the ball. This is a different technique than other softball pitches, and it explains why the ball travels so fast.

Introducing Speed

How fast or slow something moves is its speed. Speed determines how far something travels in a given amount of time. The SI unit for speed is meters per second (m/s). Speed may be constant, but often it varies from moment to moment.

Average Speed

Even if speed varies during the course of a trip, it’s easy to calculate the average speed by using this formula:

\text{speed}=\frac{\text{distance}}{\text{time}}

For example, assume you go on a car trip with your family. The total distance you travel is 120 miles, and it takes 3 hours to travel that far. The average speed for the trip is:

\text{speed} & =\frac{120 \ \text{mi}}{3 \ \text{h}}\\& =40 \ \text{mi/h}

Q: Terri rode her bike very slowly to the top of a big hill. Then she coasted back down the hill at a much faster speed. The distance from the bottom to the top of the hill is 3 kilometers. It took Terri ¼ hour to make the round trip. What was her average speed for the entire trip? (Hint: The round-trip distance is 6 km.)

A: Terri’s speed can be calculated as follows:

\text{speed} & =\frac{6 \ \text{km}}{0.25 \ \text{h}}\\& =24 \ \text{km/h}

Instantaneous Speed

When you travel by car, you usually don’t move at a constant speed. Instead you go faster or slower depending on speed limits, traffic lights, the number of vehicles on the road, and other factors. For example, you might travel 65 miles per hour on a highway but only 20 miles per hour on a city street (see the pictures in the Figure below.) You might come to a complete stop at traffic lights, slow down as you turn corners, and speed up to pass other cars. Therefore, your speed at any given instant, or your instantaneous speed, may be very different than your speed at other times. Instantaneous speed is much more difficult to calculate than average speed. 

 [1]

Cars race by in a blur of motion on an open highway but crawl at a snail’s pace when they hit city traffic.

Calculating Distance or Time from Speed

If you know the average speed of a moving object, you can calculate the distance it will travel in a given period of time or the time it will take to travel a given distance. To calculate distance from speed and time, use this version of the average speed formula given above:

distance = speed × time

For example, if a car travels at an average speed of 60 km/h for 5 hours, then the distance it travels is:

distance = 60 km/h × 5 h = 300 km

To calculate time from speed and distance, use this version of the formula:

\text{time}=\frac{\text{distance}}{\text{speed}}

Q: If you walk 6 km at an average speed of 3 km/h, how much time does it take?

A: Use the formula for time as follows:

\text{time} & =\frac{\text{distance}}{\text{speed}}\\& =\frac{6 \ \text{km}}{3 \ \text{km/h}}\\& =2 \ \text{h}

Summary

  • How fast or slow something moves is its speed. The SI unit for speed is meters per second (m/s).
  • Average speed is calculated with this formula:\text{speed}=\frac{\text{distance}}{\text{time}}
  • Speed may be constant, but often it varies from moment to moment. Speed at any given instant is called instantaneous speed. It is much more difficult to calculate than average speed.
  • Distance or time can be calculated by solving the average speed formula for distance or time.

Vocabulary

  • speed: How quickly or slowly something moves; calculated as distance divided by time.

Speed Explained

Sample Problem (Skip to 1:30 in the video)

Sample Problem (Skip to 0:35 in the video)

Image Attributions

  1. ^ Credit: Left: Kenny Louie; Right: Mario Roberto Duran Ortiz; License: CC BY-NC 3.0

Velocity


Ramey and her mom were driving down this highway at 45 miles per hour, which is the speed limit on this road. As they approached this sign, Ramey’s mom put on the brakes and started to slow down so she could safely maneuver the upcoming curves in the road. This speed limit sign actually represents two components of motion: speed and direction.

Speed and Direction

Speed tells you only how fast or slow an object is moving. It doesn’t tell you the direction the object is moving. The measure of both speed and direction is called velocity. Velocity is a vector. A vector is measurement that includes both size and direction. Vectors are often represented by arrows. When using an arrow to represent velocity, the length of the arrow stands for speed, and the way the arrow points indicates the direction. If you’re still not sure of the difference between speed and velocity, watch the cartoon.

Using Vector Arrows to Represent Velocity

The arrows in the Figure below represent the velocity of three different objects. Arrows A and B are the same length but point in different directions. They represent objects moving at the same speed but in different directions. Arrow C is shorter than arrow A or B but points in the same direction as arrow A. It represents an object moving at a slower speed than A or B but in the same direction as A.

Differences in Velocity

Objects have the same velocity only if they are moving at the same speed and in the same direction. Objects moving at different speeds, in different directions, or both have different velocities. Look again at arrows A and B from the Figure above. They represent objects that have different velocities only because they are moving in different directions. A and C represent objects that have different velocities only because they are moving at different speeds. Objects represented by B and C have different velocities because they are moving in different directions and at different speeds.

Q: Jerod is riding his bike at a constant speed. As he rides down his street he is moving from east to west. At the end of the block, he turns right and starts moving from south to north, but he’s still traveling at the same speed. Has his velocity changed?

A: Although Jerod’s speed hasn’t changed, his velocity has changed because he is moving in a different direction.

Q: How could you use vector arrows to represent Jerod’s velocity and how it changes?

A: The arrows might look like this (see Figure below):

Calculating Average Velocity

You can calculate the average velocity of a moving object that is not changing direction by dividing the distance the object travels by the time it takes to travel that distance. You would use this formula:

\text{velocity}=\frac{\text{distance}}{\text{time}}

This is the same formula that is used for calculating average speed. It represents velocity only if the answer also includes the direction that the object is traveling.

Let’s work through a sample problem. Toni’s dog is racing down the sidewalk toward the east. The dog travels 36 meters in 18 seconds before it stops running. The velocity of the dog is:

\text{velocity} & =\frac{\text{distance}}{\text{time}}\\& =\frac{36 \ \text{m}}{18 \ \text{s}}\\& =2 \ \text{m/s east}

Note that the answer is given in the SI unit for velocity, which is m/s, and it includes the direction that the dog is traveling.

Q: What would the dog’s velocity be if it ran the same distance in the opposite direction but covered the distance in 24 seconds?

A: In this case, the velocity would be:

\text{velocity} & =\frac{\text{distance}}{\text{time}}\\& =\frac{36 \ \text{m}}{24 \ \text{s}}\\& =1.5 \ \text{m/s west}

Watch the following video for a comparison of average velocity and instantaneous velocity.  You only need to watch up to 11:30, but if you watch past that time you will get a good introduction to acceleration (which we will cover next week).

Summary

  • Velocity is a measure of both speed and direction of motion. Velocity is a vector, which is a measurement that includes both size and direction.
  • Velocity can be represented by an arrow, with the length of the arrow representing speed and the way the arrow points representing direction.
  • Objects have the same velocity only if they are moving at the same speed and in the same direction. Objects moving at different speeds, in different directions, or both have different velocities.
  • The average velocity of an object moving in a constant direction is calculated with the formula: \text{velocity}=\frac{\text{distance}}{\text{time}}. The SI unit for velocity is m/s, plus the direction the object is traveling.

Vocabulary

  • vector: Measure such as velocity that includes both size and direction; may be represented by an arrow.
  • velocity: Measure of both speed and direction of motion.

Acceleration

Imagine the thrill of riding on a roller coaster like the one in Figure below. The coaster crawls to the top of the track and then flies down the other side. It also zooms around twists and turns at breakneck speeds. These changes in speed and direction are what make a roller coaster ride so exciting. Changes in speed or direction are called acceleration.

Did you ever ride on a roller coaster like this one? It’s called the "Blue Streak" for a reason. As it speeds around the track, it looks like a streak of blue.

Defining Acceleration

Acceleration is a measure of the change in velocity of a moving object. It shows how quickly velocity changes. Acceleration may reflect a change in speed, a change in direction, or both. Because acceleration includes both a size (speed) and direction, it is a vector.

People commonly think of acceleration as an increase in speed, but a decrease in speed is also acceleration. In this case, acceleration is negative. Negative acceleration may be called deceleration. A change in direction without a change in speed is acceleration as well. You can see several examples of acceleration in Figure below.

How is velocity changing in each of these pictures?

If you are accelerating, you may be able to feel the change in velocity. This is true whether you change your speed or your direction. Think about what it feels like to ride in a car. As the car speeds up, you feel as though you are being pressed against the seat. The opposite occurs when the car slows down, especially if the change in speed is sudden. You feel yourself thrust forward. If the car turns right, you feel as though you are being pushed to the left. With a left turn, you feel a push to the right. The next time you ride in a car, notice how it feels as the car accelerates in each of these ways. For an interactive simulation about acceleration, go to this URL:http://phet.colorado.edu/en/simulation/moving-man.

Calculating Acceleration

Calculating acceleration is complicated if both speed and direction are changing. It’s easier to calculate acceleration when only speed is changing. To calculate acceleration without a change in direction, you just divide the change in velocity (represented by \Delta \text{v}) by the change in time (represented by \Delta \text{t}). The formula for acceleration in this case is:

\text{Acceleration} = \frac{\Delta \text{v}}{\Delta \text{t}}

Consider this example. The cyclist in Figure below speeds up as he goes downhill on this straight trail. His velocity changes from 1 meter per second at the top of the hill to 6 meters per second at the bottom. If it takes 5 seconds for him to reach the bottom, what is his acceleration, on average, as he flies down the hill?

\text{Acceleration} = \frac{\Delta \text{v}}{\Delta \text{t}}=\frac{6 \ \text{m/s} - 1 \ \text{m/s}}{5 \ \text{s}} = \frac{5 \ \text{m/s}}{5 \ \text{s}} = \frac{1 \ \text{m/s}}{1 \ \text{m}} = 1 \ \text{m/s}^2

In words, this means that for each second the cyclist travels downhill, his velocity increases by 1 meter per second (on average). The answer to this problem is expressed in the SI unit for acceleration: \text{m/s}^2 ("meters per second squared").

Gravity helps this cyclist increase his downhill velocity.

You Try It!

Problem: Tran slowed his skateboard as he approached the street. He went from 8 m/s to 2 m/s in a period of 3 seconds. What was his acceleration?

Velocity-Time Graphs

The acceleration of an object can be represented by a velocity–time graph like the one in Figure below. A velocity-time graph shows how velocity changes over time. It is similar to a distance-time graph except the y-axis represents velocity instead of distance. The graph in Figure below represents the velocity of a sprinter on a straight track. The runner speeds up for the first 4 seconds of the race, then runs at a constant velocity for the next 3 seconds, and finally slows to a stop during the last 3 seconds of the race.

This graph shows how the velocity of a runner changes during a 10-second sprint.

In a velocity-time graph, acceleration is represented by the slope of the graph line. If the line slopes upward, like the line between A and B in Figure above, velocity is increasing, so acceleration is positive. If the line is horizontal, as it is between B and C, velocity is not changing, so acceleration is zero. If the line slopes downward, like the line between C and D, velocity is decreasing, so acceleration is negative. 

Lesson Summary

  • Acceleration is a measure of the change in velocity of a moving object. It shows how quickly velocity changes and whether the change is positive or negative. It may reflect a change in speed, a change in direction, or both.
  • To calculate acceleration without a change in direction, divide the change in velocity by the change in time.
  • The slope of a velocity-time graph represents acceleration.

Lesson Review Questions

Recall

  1. What is acceleration?
  2. How is acceleration calculated?
  3. What does the slope of a velocity-time graph represent?

Apply Concepts

  1. The velocity of a car on a straight road changes from 0 m/s to 6 m/s in 3 seconds. What is its acceleration?

Think Critically

  1. Because of the pull of gravity, a falling object accelerates at 9.8 m/s2. Create a velocity-time graph to represent this motion.

Points to Consider

Acceleration occurs when a force is applied to a moving object.

  • What is force? What are some examples of forces?
  • What forces might change the velocity of a moving object? (Hint: Read the caption to Figure above.)

Units of Acceleration

Watch the following video to learn more about units of acceleration, and how to interpret them:

Relative Motion


The wings of this hummingbird are moving so fast that they’re just a blur of motion. You can probably think of many other examples of things in motion. If you can’t, just look around you. It’s likely that you’ll see something moving, and if nothing else, your eyes will be moving. So you know from experience what motion is. No doubt it seems like a fairly simple concept. However, when you read this article, you’ll find out that it’s not quite as simple as it seems.

Defining Motion

In science, motion is defined as a change in position. An object’s position is its location. Besides the wings of the hummingbird in opening image, you can see other examples of motion in the Figure below. In each case, the position of something is changing.

 [1]

Q: In each picture in the Figure above, what is moving and how is its position changing?

A: The train and all its passengers are speeding straight down a track to the next station. The man and his bike are racing along a curving highway. The geese are flying over their wetland environment. The meteor is shooting through the atmosphere toward Earth, burning up as it goes.

Frame of Reference

There’s more to motion than objects simply changing position. You’ll see why when you consider the following example. Assume that the school bus pictured in the Figure below passes by you as you stand on the sidewalk. It’s obvious to you that the bus is moving, but what about to the children inside the bus? The bus isn’t moving relative to them, and if they look at the other children sitting on the bus, they won’t appear to be moving either. If the ride is really smooth, the children may only be able to tell that the bus is moving by looking out the window and seeing you and the trees whizzing by.

 [2]

This example shows that how we perceive motion depends on our frame of reference. Frame of reference refers to something that is not moving with respect to an observer that can be used to detect motion. For the children on the bus, if they use other children riding the bus as their frame of reference, they do not appear to be moving. But if they use objects outside the bus as their frame of reference, they can tell they are moving. The video below illustrates other examples of how frame of reference is related to motion.

Q: What is your frame of reference if you are standing on the sidewalk and see the bus go by? How can you tell that the bus is moving?

A: Your frame of reference might be the trees and other stationary objects across the street. As the bus goes by, it momentarily blocks your view of these objects, and this helps you detect the bus’ motion.

Watch the video below:


Summary

  • Motion is defined as a change of position.
  • How we perceive motion depends on our frame of reference. Frame of reference refers to something that is not moving with respect to an observer that can be used to detect motion.

Vocabulary

  • frame of reference: Something that is not moving with respect to an observer that can be used to detect motion.
  • motion: Change in position.

Review

  1. How is motion defined in science?
  2. Describe an original example that shows how frame of reference influences the perception of motion.

Image Attributions

  1. ^ Credit: Train: John H. Gray; Bike: Flickr:DieselDemon; Geese: Don McCullough; Meteor: Ed Sweeney (Flickr:Navicore); License: CC BY-NC 3.0
  2. ^ Credit: Bus: Flickr:torbakhopper; Children: Flickr:woodleywonderworks;License: CC BY-NC 3.0

Kinematic Equations

Kinematics comes from the Greek word for "motion", which means that kinematic equations are the "equations of motion".  These equations use the following symbols:

  • stands for "average velocity", and is measured in m/s
  • vi or stands for "initial velocity", and is measured in m/s.  Sometimes they also use the symbol v0, which means "velocity at time zero".  They're actually the same thing.
  • vf stands for "final velocity", and is measured in m/s
  • d stands for "distance", and is measured in meters
  • t stands for "time", and is measured in seconds
  • a stands for "acceleration", and is measured in m/s^2

The following kinematic equations use these symbols to describe motion:



These equations apply to situations with uniform acceleration, which means that your acceleration does not change.  Watch the following videos for a few examples of how to use these equations.







Tips and Tricks:

  • When deciding what equation to use, make a list of the variables given to you in the problem, and the variable you want to find out.  Then use the equation that includes all of those symbols.
  • Check to make sure all of your units match up.  All distance units should match and all time units should match.
  • Follow order of operations carefully as you work with these equations.

Acceleration Due to Gravity


In the absence of air resistance, all objects fall toward the earth with the same acceleration.  Man, however, make maximum use of air resistance in the construction of parachutes for both entertainment and military use.

The image at left was taken during a 2008 Graduation demonstration jump by the U.S. Army Parachute Team.  The 2008 team contained the first amputee member and the largest number of females in history.

Concepts of Falling Objects

Acceleration Due to Gravity

One of the most common examples of uniformly accelerated motion is that of an object allowed to fall vertically to the earth.  In treating falling objects as uniformly accelerated motion, we must ignore air resistance.  Galileo’s original statement about the motion of falling objects is:

At a given location on the earth and in the absence of air resistance, all objects fall with the same uniform acceleration.

We call this acceleration due to gravity on the earth and we give it the symbol g.  The value of gis 9.80 m/s2.  All of the equations involving constant acceleration can be used for falling bodies but we insert g wherever “a” appeared and the value of g is always 9.80 m/s2. 

Example:  A rock is dropped from a tower 70.0 m high.  How far will the rock have fallen after 1.00 s, 2.00 s, and 3.00 s?  Assume the distance is positive downward.

Solution:  We are looking for displacement and we have time and acceleration.  Therefore, we can use d = \frac{1}{2} at^2.

Displacement after 1.00 \ \text{s} = \left(\frac{1}{2}\right)(9.80 \ \text{m/s}^2)(1.00 \ \text{s})^2 = 4.90 \ \text{m}

Displacement after 2.00 \ \text{s} = \left(\frac{1}{2}\right)(9.80 \ \text{m/s}^2)(2.00 \ \text{s})^2 = 19.6 \ \text{m}

Displacement after 3.00 \ \text{s} = \left(\frac{1}{2}\right)(9.80 \ \text{m/s}^2)(3.00 \ \text{s})^2 = 44.1 \ \text{m}

Example:  (a) A person throws a ball upward into the air with an initial velocity of 15.0 m/s.  How high will it go before it comes to rest?  (b)  How long will the ball be in the air before it returns to the person’s hand?

Solution:  In part (a), we know the initial velocity (15.0 m/s), the final velocity (0 m/s), and the acceleration -9.80 m/s2.  We wish to solve for the displacement, so we can use v{_f}^2 = v{_i}^2 + 2adand solve for d.

d=\frac{v{_f}^2-v{_i}^2}{2a}=\frac{(0 \ \text{m/s})^2-(15.0 \ \text{m/s})^2}{(2)(9.80 \ \text{m/s}^2)}=11.5 \ \text{m}

There are a number of methods by which we can solve part (b).  Probably the easiest is to divide the distance traveled by the average velocity to get the time going up and then double this number since the motion is symmetrical – that is, time going up equals the time going down.

The average velocity is half of 15.0 m/s or 7.5 m/s and dividing this into the distance of 11.5 m yields 1.53 seconds.  This is the time required for the ball to go up and the time for the ball to come down will also be 1.53 s, so the total time for the trip up and down is 3.06 seconds.

Example:   A car accelerates with uniform acceleration from 11.1 m/s to 22.2 m/s in 5.0 s.  (a) What was the acceleration and (b) how far did it travel during the acceleration?

Solution: 

(a)  a=\frac{\Delta v}{\Delta t}=\frac{22.2 \ \text{m/s}-11.1 \ \text{m/s}}{5.0 \ \text{s}}=2.22 \ \text{m/s}^2

(b)  We can find the distance traveled by d = v_it + \frac{1}{2} at^2 and we can also find the distance traveled by determining the average velocity and multiply it by the time.

d &= v_it + \frac{1}{2} at^2 \\&= (11.1 \ \text{m/s})(5.0 \ \text{s}) + \left(\frac{1}{2}\right)(2.22 \ \text{m/s}^2)(5.0 \ \text{s})^2 \\&= 55.5 \ \text{m} + 27.8 \ \text{m} \\&= 83 \ \text{m}

d = (v_{\text{ave}})(t) = (16.6 \ \text{m/s})(5.0 \ \text{s}) = 83 \ \text{m}

Example:  A stone is dropped from the top of a cliff.  It is seen to hit the ground after 5.5 s.  How high is the cliff?

Solution: 

d = v_it + \frac{1}{2} at^2 = (0 \ \text{m/s})(5.5 \ \text{s}) + \left(\frac{1}{2}\right)(9.80 \ \text{m/s}^2)(5.5 \ \text{s})^2 = 150 \ \text{m}

Summary

  • At a given location on the earth and in the absence of air resistance, all objects fall with the same uniform acceleration.
  • We call this acceleration the acceleration due to gravity on the earth and we give it the symbol g. 
  • The value of g is 9.80 m/s2.

Common Misconceptions

Practice

This url shows a video of a discussion and demonstration of the acceleration due to gravity.

Position vs. Time Graphs

Introduction to Graphing

Position vs Time with No Acceleration

Position vs Time with Positive Acceleration

Position vs Time with Negative Acceleration

Graphing Distance and Time

The motion of an object can be represented by a distance-time graph like Graph 1 in the Figurebelow. In this type of graph, the y-axis represents distance and the x-axis represents time. A distance-time graph shows how far an object has traveled at any given time since it started moving. However, it doesn’t show the direction(s) the object has traveled.  

Q: In the Figure above, what distance has the object traveled by the time 5 seconds have elapsed?

A: The object has traveled a distance of about 25 meters.

Slope Equals Speed

In a distance-time graph, the speed of the object is represented by the slope, or steepness, of the graph line. If the graph line is horizontal, like line B in Graph 2 in the Figure below, then the slope is zero and so is the speed. In other words, the object is not moving. The steeper the line is, the greater the slope of the line is and the faster the object is moving.

Q: In graph 2, which line represents a faster speed: line A or line C?

A: Line A represents a faster speed because it has a steeper slope.

Calculating Average Speed from a Distance-Time Graph

It’s easy to calculate the average speed of a moving object from a distance-time graph. Average speed equals a change in distance (represented by Δd) divided by the corresponding change in time (represented by Δt):

\text{speed}= \frac{\Delta d}{\Delta t}\\

For example, in Graph 3 in the Figure below, the average speed between 1 second and 4 seconds is:

\text{speed} & = \frac{\Delta d}{\Delta t}\\& =\frac{3 \ \text{m}-2 \ \text{m}}{4 \ \text{s}-1 \ \text{s}}\\& =\frac{1 \ \text{m}}{3 \ \text{s}}\\& =0.3 \ \text{m/s}

Q: In graph 3, what is the average speed between 0 and 4 seconds?

A: The average speed is:

\text{speed} & = \frac{\Delta d}{\Delta t}\\& =\frac{3 \ \text{m}-0 \ \text{m}}{4 \ \text{s}-0 \ \text{s}}\\& =\frac{3 \ \text{m}}{4 \ \text{s}}\\& =0.8 \ \text{m/s}

Summary

  • Motion can be represented by a distance-time graph, which plots distance on the y-axis and time on the x-axis.
  • The slope of a distance-time graph represents speed. The steeper the slope is, the faster the speed.
  • Average speed can be calculated from a distance-time graph as the change in distance divided by the corresponding change in time.

Review

  1. Describe how to make a distance-time graph.
  2. What is the slope of a line graph? What does the slope of a distance-time graph represent?
  3. Can a line on a distance-time graph have a negative slope, that is, can it slope downward from left to right? Why or why not?
  4. In graph 1 above, the speed of the object is constant. What is the object’s speed in m/s?
  5. In graph 3 above, describe the motion of the object at the time of 2 seconds.

Velocity vs. Time Graphs


The sprinter in this image is just taking off from the starting blocks to run a short race down a straight track. She starts in a burst of speed and will pick up even more speed during the first few seconds of the race. She’ll keep running at top speed until she crosses the finish line. Only then will she slow down. Velocity is a measure of both speed and direction of motion. A change in velocity is called acceleration. In the case of the sprinter, she accelerates as she runs down the track because her speed is changing even though her direction stays the same.

Plotting Velocity Against Time

The changing velocity of the sprinter—or of any other moving person or object—can be represented by a velocity-time graph like the one in the Figure below for the sprinter. A velocity-time graph shows how velocity changes over time. The sprinter’s velocity increases for the first 4 seconds of the race, it remains constant for the next 3 seconds, and it decreases during the last 3 seconds after she crosses the finish line.

Acceleration and Slope

In a velocity-time graph, acceleration is represented by the slope, or steepness, of the graph line. If the line slopes upward, like the line between 0 and 4 seconds in the graph above, velocity is increasing, so acceleration is positive. If the line is horizontal, as it is between 4 and 7 seconds, velocity is constant and acceleration is zero. If the line slopes downward, like the line between 7 and 10 seconds, velocity is decreasing and acceleration is negative. Negative acceleration is called deceleration.

Q: Assume that another sprinter is running the same race. The other runner reaches a top velocity of 9 m/s by 4 seconds after the start of the race. How would the first 4 seconds of the velocity-time graph for this runner be different from the graph above?

A: The graph line for this runner during seconds 0–4 would be steeper (have a greater slope). This would show that acceleration is greater during this time period for the other sprinter.

Summary

  • A velocity-time graph shows changes in velocity of a moving object over time.
  • The slope of a velocity-time graph represents acceleration of the moving object.

Vocabulary

  • velocity: Measure of both speed and direction of motion.

Review

  1. Describe a velocity-time graph. What does the slope of the graph line represents?
  2. In the velocity-time graph above, the sprinter reaches a velocity of 2 m/s in just 1 second. At a constant rate of acceleration, how long does it take for her to double this velocity? What is her acceleration during this time period?
  3. Create a velocity-time graph by plotting the data in the Table below.
Velocity (m/s)Time (s)
10 1
30 2
50 3
40 4
40 5