Lab 4

Motion

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Objectives

This lab exercise is designed to help you visualize the RELATIONSHIPS between the variables which we use to describe motion. You may wish to consult chapters 5 and 6 in the Booth and Bloom text and Program 12 for reference.

1. Describe motion in terms of distance and time. 

2. Distinguish between velocity and acceleration. 

3. Recognize the RELATIONSHIPS between numbers, equations, and graphs. 

4. Visualize the concepts of velocity and acceleration graphically. 

5. Understand the concepts of ratio and rate of change. 

6. Understand the RELATIONSHIPS between distance and time for constant and uniformly accelerated motion. 

7. Understand the meaning of the slope and area of a graph of velocity vs. time

References

Program 12

Spielberg & Anderson pp. 60-71

Booth & Bloom pp. 83-85, 97-99

Introduction

Galileo was the first to consider motion as a quantitative RELATIONSHIP between distance and time. Prior to Galileo's analysis the concept of speed was nothing more than intuitive.

 It is easy to decide which of two things is moving faster if they are running a race. The faster horse will run the course in a shorter time than the slower horse.

 It is another thing altogether to describe in general terms exactly what the difference is between a fast and a slow moving object.

The idea that the RELATIONSHIP between distance and time can be expressed as a ratio was Galileo's. He reasoned that he larger the ratio, the faster the speed. There is a number which represents the concept of speed such that the motion of two objects can be compared even when they are not moving in the same vicinity at the same time. We take this for granted today, but it is really quite a nifty idea.

The idea that RELATIONSHIPS such as ratios can be displayed pictorially is usually attributed to Descartes, from whom we get the name, Cartesian coordinates. Displaying RELATIONSHIPS in this way allows for a quicker and more intuitive understanding of them .

 The concept of a rate of change is a fairly sophisticated one. It is no coincidence that the words"rate" and "ratio" are similar. They essentially mean the same thing, to compare one number with another. In the physical world the numbers we are comparing are measurable physical quantities. The RELATIONSHIPS are the important thing, not the numbers themselves.

 In this exercise we will try to visualize and understand motion and the RELATIONSHIPS between numbers, ratios, and graphical representations of motion.

A. Velocity as a RATIO of distance and time

Velocity is the rate of change of location with time. When location is measured from a reference point in a particular direction then we can think of it as distance.

Procedure A1. Without actually calculating the speed, determine which pair of numbers in each set in table 1 represents the highest velocity. Try to release your mind from the paradigm of calculation and look at the RELATIONSHIP.

Highest velocity is the greatest distance in the smallest amount of time. For each of the two sets, mark the box of the greatest velocity.

Procedure A2. For each the pairs of numbers in table 2, determine which represents the greatest speed by calculating the ratio d/t. Write the calculated numbers in the appropriate blanks. Note that the units of velocity are units of distance divided by units of time.

Procedure A3. Plot the number pairs from table 2 on a graph. Plot time on the horizontal (x) axis. Then draw a straight line from the origin of the graph (the point 0,0) to each of the plotted points.

Procedure A Questions

1. Which point has the line with the steepest slope on the graph? Which point has the line with smallest slope? 

2. Which points lie on the same straight line? 

3. Using the graph, state a general RELATIONSHIP between speed and the steepness (slope) of the line on the graph?

B. Acceleration as a RATIO of velocity and time

When velocity changes we say that acceleration has occurred. If velocity changes at a constant rate we say the acceleration is uniform. For uniform acceleration the RELATIONSHIPS between distance and time show a remarkable symmetry when viewed graphically. The units of acceleration are units of velocity divided by units of time.

Procedure B. Table 3 represents the velocity as read from a car's speedometer at certain times. For each pair of numbers determine the ratio of v/t and complete the table. Then answer the questions below.

Procedure B Questions

1. In what units would you express the acceleration for this example?

2. Is the rate of change of velocity constant? Is the acceleration uniform? 

3. Fill in the blanks with the appropriate number:

In table 3 it is always true that one can determine v by multiplying t by the number______. In equation form this would be written v = ___ t

C. Distance and time in uniformly accelerated motion.

The RELATIONSHIPS between distance traveled and time elapsed seem complicated when considering uniformly accelerated motion. With the help of a graph these RELATIONSHIPS seem much simpler. That is why we use a graph in the first place!

Procedure C1. The numbers in table 4 represent the distance traveled in equal time intervals by a car which starts from rest and accelerates uniformly.. Plot the values of time and distance from table 4 on the graph. Plot time on the horizontal (x) axis. Ignore the velocity and time squared for now. You will return to this data later.

Procedure C2. On the graph below draw a vertical line from each point to the time axis. Then draw a horizontal line from each point to the distance axis. This is done for the point (3,18) as an example.

1. Lay a straightedge on the graph. Does this data plot as a straight line?

2. The vertical lines are equally spaced but the horizontal lines are not. State the physical meaning of this in words.

3. Does the distance traveled during each interval of time change in a regular way? If so describe it here. If you're not sure, skip this question and go on.

Procedure C3. From table 4 determine the distance traveled during certain time intervals.

1. Write the answer in the row labeled, intervals {2 - 1} in table 5 below. (Find the values for t₂ - t₁ and d₂ - d₁). These reprsent the distance traveled between the times in the first column.

Example: for distance is the distance traveled during the time from 5 seconds to seven seconds. To calculate it subtract the distance at 7 seconds from the distance at 5 seconds: d₇ - d₅ = 98 - 50 = 48

2. Subtract the time as indicated and write those numbers in the table.

3. Do the same for the intervals {4 -2} and {6 - 3}.

Procedure C4. Pythagorean numerology! What patterns or RELATIONSHIP exist between the numbers representing the time and distance intervals in table 5?

A. Finding the Patterns

1. Divide distance by time for each interval in table 5 and see if there is a pattern. You might find it easier to see this pattern if you leave the ratios as fractions and reduce them.

B. Organizing the patterns.

1. In table 6 determine the ratios as indicated for the given intervals. Use the numbers from Table 5. The numbers in the top and bottom of the fraction: are the numbers that represent the times and distances calculated in table 5.

If you've done everything right, the numbers in the time squared column in table 6 should be exactly the same as the numbers in the distance column.

When the time is doubled (twice as much time, e.g. between 1 and 2 seconds, between 2 and 4 seconds, and between 3 and 6 seconds, The distance traveled is ALWAYS four times greater than the distance traveled in those two time units. Why? Because two squared is four and distance is proportional to the time dquared.

This is not magic, or a trick of numbers. It is not a Pythagorean thing that results from the properties of the numbers themselves.

It indicates the type of RELATIONSHIP between the time and the distance for uniform acceleration.

Uniform acceleration is a special kind of motion because this RELATIONSHIP exists between time and distance.

Distance is directly proportional to the square of time in uniformly accelerated motion. An object that is uniformly accelerating travels four times the distance in twice the time.

This is true for any time intervals when distance and time are compared.

Try it for {8 - 2}! This is four times the time interval, eight seconds compared to 2 seconds: (8/2 = 4)

This is what we mean by: distance is proportional to time squared?

Procedure C5. Plot a graph of distance vs. time squared.

1. Using the data from table 4, square the time values from the data table. In other words, multiply each of the times by itself and write the results in the spaces in table 4.

2. Plot the values of distance and the square of time on the blank graph. Use a straightedge to connect the points. They should plot in a straight line because distance is directly proportional to the square of time when acceleration is uniform.

3. Determine the numerical slope of this straight line according to the illustration which follows. You may want to review the concept of the slope from lab exercise 3.

D. Putting it together

What you have been doing is looking at the numerical RELATIONSHIPS between distance and time for objects in motion, either at constant speed, or uniformly accelerated. The numerical "magic" is due to the fact that there are RELATIONSHIPS between distance and time which characterize motion. 

These RELATIONSHIPS can be described in words, in pictures, or in equation. All three representations give a slightly different way of visualizing these RELATIONSHIPS. 

Here's a way to see all of these RELATIONSHIPS. Let's define the terms used:

Using the properties of numbers and shapes we can visualize these physical RELATIONSHIPS graphically. 

Here is table 4 again. By now you should be able to predict what a graph of velocity vs. time will look like. If you cannot, take a few minutes to draw the graph.

You will note that the velocity increases 4 m/s each second.

4 m/s is added to the velocity each second, so the velocity increases at a rate of 4 meters per second every second (this is the same as meters per second per second and it written m/s².

This represents the slope of the graph. Writing the equation v = 4t expresses the same RELATIONSHIP. Here we are using the general RELATIONSHIP v=at where a=4.

This can also help to visualize the RELATIONSHIP between distance and the square of time in part C. The mathematical logic is shown below. If you have difficulty following this, imagine how difficult it must have been for Galileo and his contemporaries without the benefit of the notation of modern algebra! You might find some help in Chapters 5 & 6 of the Booth and Bloom text and in Program 12. Note that time appears twice (once from the velocity, once from the acceleration).

Procedure D Questions

1. State the relationship between distance and time for uniformly accelerated motion, and the relationship between velocity and time.

2. Would the graph of velocity vs. time in table 4 be a straight line, or a curved line? 

3. What would be the numerical value and units of the slope of the graph? 

4. What physical quantity would the slope represent in the graph from question 1?

Summary

Here is what we have learned in this exercise:

1. Velocity depends on both distance and time. 

2. Constant velocity plots as a straight line on a distance vs. time graph. 

3. The steepness of the line on the distance vs. time graph is a measure of velocity. 

4. Acceleration is the addition of velocity with elapsed time. 

5. Uniform acceleration means that the velocity increases by the same amount in each time interval. 

6. Accelerated motion plots as a curved line on a distance vs. time graph. 

7. Increasingly greater distances are traveled in equal time intervals in accelerated motion. 

8. Uniformly accelerated motion plots as a straight line on a velocity vs. time graph. 

9. The slope of a graph of velocity vs. time represents acceleration while the area of the graph represents the distance traveled. 

This may seem like a lot of material to assimilate all at once, but it is not really difficult. It may take some time and some thought to put it all together to fully understand accelerated motion. Hopefully it gives you an idea of the way in which RELATIONSHIPS, between numbers, geometry, and measurable quantities can be visualized. It is the basis of most of our understanding of the physical world.