Honors Physics - Chapter 2: Lesson 4

Chapter 2: Lesson 4

Graphing

Chapter 2: Lesson 4 - Graphing

Motion Diagrams:

Physicists are lazy. They don't want to use a bunch of words to describe various situations. They would rather glace at a picture or a graph to portray what is happening in a situation. Below are two ways that we can use pictures to describe motion.

Motion Diagram

A series of images showing the positions of a moving object at equal time intervals is called a motion diagram.

Particle Model

A particle model is a simplified version of a motion diagram in which the object in motion is replaced by a series of single points.

Look at the motion diagram (picture a.) You can see that the man is running to the right (the direction.) You can also see that the spacing in between each picture of the man is equal (the magnitude.) This means that the man is running with a constant velocity. It takes a lot of time to draw the man each time, so physicists want to make that even easier and use a dot to represent the object. In this case, the man.

The particle model (picture b) shows both the magnitude and the direction of the object. The spacing between the dots are the same (the magnitude) and the arrow shows that the object is moving to the right (the direction.) It is much easier to draw a dot than it is an entire man!

How to Calculate Slope:

When looking at a graph, you can find the slope of the line by this equation. We can abbreviate slope with a lowercase m. The y axis is the vertical (up and down) axis. The x axis is the horizontal (left and right) axis. Remember the ∆y stands for change in y position. Anytime you see the delta, ∆, you subtract the initital quantity from the final quantity, yf-yi. The same goes for the x-axis.

When chosing what two points to pick on a graph, you want to include as much of your graph as possible. In science, we want to pick two points that are as far apart as possible. I know in math, it is great to pick the (0, 0) point because it makes the calculation easy, but if you didn't physicially gather that (0, 0) data point in the lab, you cannot use it as a valid data point. To get more a more accurate calculation of your slope, it is best to pick the next point if that point is valid.

Distance vs Time Graphs:

Look at the spacing between these cars on this motion diagram.

The spacing between the cars is the same, so that tells us that the motion is the same. This car is moving with a constant, rightward (+) velocity of 10 m/s.

Here is the position time graph that represents the motion diagram. Look at the spacing between the points on the graph. The spacing is the same, so that tells us that the motion is the same.

Looking at the spacing between the points on a position or distance vs time graph can help us describe the motion.

On any graph, the dependent variable is always on the y-axis and the independent variable is on the x-axis. For the position vs time graph, time is ALWAYS on the x-axis because time is always independent. It does not depend on what is happening around it. There is always 60 seconds in every minute, 60 minutes in every hour, and 24 hours in a day, regardless of what you are doing. Anytime you are graphing time, it will always be on the x-axis.

Position is dependent on the time. Your position during the day depends on time. In school, at 7:41 A.M., you are in first hour. At 2:25 P.M., you are going home. Position is the dependent variable and will be on the y-axis.

Look at the spacing between these cars on this motion diagram.

The spacing between the cars is getting larger with each second that passes. This tells us that the car is moving with a CHANGING, rightward (+) velocity. The car is moving to the right and speeding up (accelerating.)

Look at the spacing betwee the dots on the position time graph that represents the motion diagram. The spacing between the dots is also increasing. The car is moving with a increasing, positive velocity.

When an object is stationary, its position is staying the same over a period of time. The position time graph of an object that is not moving is a horizontal line. The object in this graph is staying at 3 m for all five seconds.

Check Your Understanding:

For the four questions below, think about the question, then click on the down arrow when you have your answer to check to see if you are correct.

What is the equation to find slope?

m=∆y/∆x

2. What is the equation to find velocity?

v=d/t

3. What is the variable on the y-axis on a distance-time graph?

Distance

4. What is the variable on the x-axis on a distance-time graph?

Time

Slope of a Distance vs Time Graph:

In the questions above, you answered some important topics. You said that slope is ∆y/∆x. You said that v=d/t and that distance is on the y-axis and time is on the x-axis.

If you look at the equation, you can see that m=∆y/∆x. ∆y is in the numerator, and since distance is on the y-axis, we will place d in the numerator as well. ∆x is in the denominator and time is on the x-axis, we will place t in the denominator as well.

You can now see that the slope of a distance time graph is d/t, which is velocity.

Instantaneous Velocity:

If you remember from lesson 2, instaneaneous velocity is the velocity at any instant.

You can find the instantaneous velocity on a distance-time graph by taking the slope of the tangent line. A tangent line touches a line in just one point, but does not intersect it.

Constant Velocity vs Acceleration:

Graphs of motion that show constant velocity look different from graphs that show a changing velocity.

This graph shows a constant velocity in a positive direction. Notice that it is a straight diagonal line.

This graph shows a changing velocity (acceleration) in a positive direction. Notice that the line is curved because the spacing in between the points increases.

Moving Right or Left and Moving Fast or Slow:

From lesson 1, we learned that moving to the right is a positive direction and moving to the left is a negative direction. From this lesson, we learned that the slope of a distance time graph is velocity. When the slope of the line is positive, the object is moving to the right, a positive direction. When the slope of the line is negative, the object is moving to the left, a negative direction.

Objects that travel faster have a steeper slope. They cover more distance is less time.

Lets look at four examples. The first two graphs both have a positive velocity and are moving to the right. The last two graphs have a negative velocity and are moving to the left. See if you can see the difference in the steepness of the slope to see which graph protrays an object that is moving faster.

The graph shows a slow, rightward (positive) constant velocity.

This graph is steeper than the first graph. This graph shows a fast rightward (positive) constant velocity.

This graph shows a slow, leftward (negative) constant velocity.

This graph is steeper than the third graph. This graph shows a fast leftward (negative) constant velocity.

In summary:

Here is a good graphic to reference when looking at the different possibilites of motion when graphed on a position or distance vs time graph.

Check Your Understanding:

For the three questions below, think about the question, then click on the down arrow when you have your answer to check to see if you are correct. Refer to the

In the first position time graph, which runner is faster? Runner A or runner B? Why?

Runner B is faster. Runner B has a steeper slope and caught up to runner A even though runner B started 50 m behind runner A. Runner B crossed the starting line at the 10 second mark.

2. In the first position time graph, when and where does runner B pass runner A?

At 45 seconds around the 190 m mark.

3. Which car (blue or red) has the greater velocity? How can you tell?

The red car has the greater velocity. The slope of the red line is larger than the blue line. The red car starts at the 4 second mark and catches up with the blue car at 7 seconds. After 7 seconds the red car is ahead of the blue car.

You have finished Lesson 4!

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