Which Statements Can Be Supported by Using a Position-time Graph? Check All That Apply.

Department Learning Objectives

By the end of this section, you will be able to practise the post-obit:

• Explain the significant of slope in position vs. time graphs
• Solve bug using position vs. fourth dimension graphs

Teacher Support

Teacher Support

The learning objectives in this section will help your students chief the post-obit standards:

• (4) Science concepts. The pupil knows and applies the laws governing motion in a variety of situations. The student is expected to:
  • (A) generate and interpret graphs and charts describing different types of motion, including the utilize of existent-fourth dimension technology such as motion detectors or photogates.

Section Fundamental Terms

dependent variable

contained variable

tangent

Instructor Support

Teacher Support

[BL] [OL] Draw a scenario, for example, in which you launch a water rocket into the air. It goes up 150 ft, stops, and then falls back to the world. Have the students assess the state of affairs. Where would they put their zip? What is the positive direction, and what is the negative direction? Have a student draw a film of the scenario on the board. Then draw a position vs. time graph describing the motion. Accept students help you consummate the graph. Is the line straight? Is information technology curved? Does it modify direction? What can they tell by looking at the graph?

[AL] Once the students take looked at and analyzed the graph, encounter if they can describe different scenarios in which the lines would be direct instead of curved? Where the lines would be discontinuous?

Graphing Position as a Office of Fourth dimension

A graph, like a picture, is worth a yard words. Graphs not only contain numerical information, they besides reveal relationships between concrete quantities. In this section, we volition investigate kinematics by analyzing graphs of position over time.

Graphs in this text have perpendicular axes, i horizontal and the other vertical. When two physical quantities are plotted against each other, the horizontal axis is unremarkably considered the independent variable, and the vertical axis is the dependent variable. In algebra, you would accept referred to the horizontal axis every bit the ten-axis and the vertical axis equally the y-axis. As in Figure 2.10, a directly-line graph has the full general form $y=\mathrm{one\; thousand}\mathrm{10}+b$ .

Hither m is the slope, defined as the rise divided by the run (every bit seen in the figure) of the straight line. The letter of the alphabet b is the y-intercept which is the indicate at which the line crosses the vertical, y-axis. In terms of a physical situation in the real world, these quantities will take on a specific significance, as we volition run across below. (Figure 2.10.)

Figure 2.ten The diagram shows a straight-line graph. The equation for the straight line is y equals mx + b.

In physics, time is usually the contained variable. Other quantities, such as deportation, are said to depend upon it. A graph of position versus time, therefore, would take position on the vertical axis (dependent variable) and time on the horizontal axis (independent variable). In this case, to what would the slope and y-intercept refer? Allow's look back at our original example when studying distance and displacement.

The drive to schoolhouse was five km from domicile. Permit's presume information technology took 10 minutes to make the drive and that your parent was driving at a constant velocity the whole time. The position versus fourth dimension graph for this section of the trip would look similar that shown in Figure 2.11.

Figure 2.11 A graph of position versus time for the drive to school is shown. What would the graph look similar if we added the render trip?

As we said before, d ₀ = 0 because nosotros phone call dwelling house our O and start computing from there. In Figure 2.11, the line starts at d = 0, as well. This is the b in our equation for a straight line. Our initial position in a position versus time graph is ever the place where the graph crosses the ten-axis at t = 0. What is the slope? The rising is the change in position, (i.due east., displacement) and the run is the change in time. This relationship can too be written

This human relationship was how we divers average velocity. Therefore, the gradient in a d versus t graph, is the average velocity.

Tips For Success

Sometimes, every bit is the instance where nosotros graph both the trip to school and the render trip, the behavior of the graph looks different during dissimilar time intervals. If the graph looks like a series of straight lines, then you can calculate the average velocity for each time interval by looking at the gradient. If y'all then want to calculate the average velocity for the entire trip, you lot can do a weighted average.

Let'due south look at another instance. Effigy two.12 shows a graph of position versus time for a jet-powered car on a very flat dry lake bed in Nevada.

Figure 2.12 The diagram shows a graph of position versus time for a jet-powered automobile on the Bonneville Table salt Flats.

Using the relationship between dependent and contained variables, we see that the slope in the graph in Figure two.12 is average velocity, five _avg and the intercept is displacement at time nix—that is, d ₀. Substituting these symbols into y = mx + b gives

or

$$d=d_0+vt\text{.}$$

2.6

Thus a graph of position versus fourth dimension gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical data about a specific situation. From the figure we can see that the car has a position of 400 one thousand at t = 0 southward, 650 m at t = 1.0 s, and so on. And nosotros tin can learn about the object'due south velocity, likewise.

Teacher Back up

Teacher Support

Teacher Demonstration

Assist students larn what different graphs of displacement vs. time wait like.

[Visual] Prepare up a meter stick.

1. If you can observe a remote control auto, have one student record times as you send the machine forward along the stick, so backwards, and so forward again with a constant velocity.
2. Take the recorded times and the change in position and put them together.
3. Go the students to omnibus yous to draw a position vs. time graph.

Each leg of the journey should be a straight line with a different slope. The parts where the car was going forward should have a positive slope. The office where information technology is going backwards would accept a negative slope.

[OL] Ask if the identify that they take every bit aught affects the graph.

[AL] Is it realistic to draw any position graph that starts at rest without some curve in it? Why might we be able to fail the bend in some scenarios?

[All] Discuss what can be uncovered from this graph. Students should exist able to read the net deportation, only they can also use the graph to make up one's mind the total distance traveled. And so ask how the speed or velocity is reflected in this graph. Direct students in seeing that the steepness of the line (slope) is a measure of the speed and that the direction of the gradient is the direction of the motion.

[AL] Some students might recognize that a curve in the line represents a sort of slope of the gradient, a preview of acceleration which they will learn almost in the next chapter.

Snap Lab

Graphing Motion

In this activity, yous will release a ball downwardly a ramp and graph the ball's displacement vs. time.

• Choose an open location with lots of space to spread out so there is less adventure for tripping or falling due to rolling assurance.

• i ball
• 1 lath
• 2 or 3 books
• 1 stopwatch
• 1 tape mensurate
• half-dozen pieces of masking tape
• 1 piece of graph paper
• 1 pencil

Procedure

1. Build a ramp past placing one finish of the board on height of the stack of books. Adjust location, as necessary, until there is no obstacle forth the straight line path from the bottom of the ramp until at least the adjacent three k.
2. Mark distances of 0.5 m, 1.0 1000, one.five m, ii.0 k, 2.five g, and iii.0 m from the bottom of the ramp. Write the distances on the record.
3. Take one person take the part of the experimenter. This person will release the ball from the tiptop of the ramp. If the ball does not achieve the 3.0 m marker, then increase the incline of the ramp by adding some other volume. Repeat this Pace equally necessary.
4. Take the experimenter release the ball. Have a second person, the timer, brainstorm timing the trial once the ball reaches the bottom of the ramp and stop the timing once the ball reaches 0.v m. Have a third person, the recorder, tape the time in a information table.
5. Echo Stride four, stopping the times at the distances of 1.0 m, 1.v chiliad, two.0 grand, two.v m, and three.0 thousand from the lesser of the ramp.
6. Apply your measurements of fourth dimension and the displacement to brand a position vs. time graph of the ball'due south motion.
7. Repeat Steps 4 through 6, with dissimilar people taking on the roles of experimenter, timer, and recorder. Practise you get the same measurement values regardless of who releases the ball, measures the time, or records the upshot? Discuss possible causes of discrepancies, if any.

Truthful or False: The average speed of the ball will be less than the boilerplate velocity of the ball.

1. Truthful

2. Simulated

Teacher Back up

Instructor Support

[BL] [OL] Emphasize that the motion in this lab is the motion of the brawl every bit it rolls along the floor. Ask students where there zero should be.

[AL] Enquire students what the graph would look like if they began timing at the tiptop versus the bottom of the ramp. Why would the graph await dissimilar? What might account for the difference?

[BL] [OL] Have the students compare the graphs fabricated with different individuals taking on unlike roles. Ask them to determine and compare average speeds for each interval. What were the absolute differences in speeds, and what were the percentage differences? Exercise the differences appear to exist random, or are there systematic differences? Why might there be systematic differences between the two sets of measurements with different individuals in each office?

[BL] [OL] Have the students compare the graphs made with different individuals taking on different roles. Ask them to decide and compare average speeds for each interval. What were the absolute differences in speeds, and what were the percent differences? Do the differences appear to exist random, or are there systematic differences? Why might there be systematic differences between the 2 sets of measurements with different individuals in each role?

Solving Problems Using Position vs. Fourth dimension Graphs

So how practise we use graphs to solve for things we desire to know like velocity?

Worked Example

Using Position–Time Graph to Calculate Average Velocity: Jet Car

Discover the boilerplate velocity of the car whose position is graphed in Figure 1.13.

Strategy

The slope of a graph of d vs. t is average velocity, since gradient equals ascension over run.

$$\text{slope =}\frac{\text{Δ}d}{\text{Δ}t}=v$$

two.7

Since the slope is constant here, any 2 points on the graph can be used to discover the slope.

Discussion

This is an impressively high land speed (900 km/h, or about 560 mi/h): much greater than the typical highway speed limit of 27 g/s or 96 km/h, but considerably shy of the record of 343 m/s or 1,234 km/h, set up in 1997.

Teacher Support

Teacher Support

If the graph of position is a straight line, and so the merely matter students demand to know to calculate the average velocity is the slope of the line, rise/run. They tin can use whichever points on the line are near user-friendly.

But what if the graph of the position is more complicated than a directly line? What if the object speeds upwardly or turns around and goes backward? Can we figure out anything about its velocity from a graph of that kind of motion? Let's take another look at the jet-powered car. The graph in Figure 2.thirteen shows its move every bit information technology is getting up to speed afterwards starting at rest. Fourth dimension starts at zero for this move (equally if measured with a stopwatch), and the deportation and velocity are initially 200 chiliad and 15 yard/s, respectively.

Figure 2.13 The diagram shows a graph of the position of a jet-powered car during the time span when information technology is speeding up. The slope of a distance versus time graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that indicate.

Figure two.14 A U.Southward. Air Force jet auto speeds down a track. (Matt Trostle, Flickr)

The graph of position versus time in Effigy 2.13 is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any indicate on a position-versus-time graph is the instantaneous velocity at that indicate. Information technology is constitute by drawing a directly line tangent to the curve at the point of involvement and taking the slope of this direct line. Tangent lines are shown for two points in Effigy 2.13. The average velocity is the net deportation divided by the fourth dimension traveled.

Worked Example

Using Position–Time Graph to Calculate Average Velocity: Jet Machine, Accept Two

Calculate the instantaneous velocity of the jet car at a time of 25 due south by finding the slope of the tangent line at indicate Q in Figure 2.13.

Strategy

The slope of a bend at a indicate is equal to the gradient of a straight line tangent to the bend at that bespeak.

Discussion

The entire graph of v versus t tin can be obtained in this way.

Teacher Support

Instructor Support

A curved line is a more than complicated case. Define tangent as a line that touches a curve at only 1 point. Prove that as a directly line changes its bending next to a curve, it actually hits the curve multiple times at the base of operations, merely only one line will never touch at all. This line forms a right angle to the radius of curvature, but at this level, they can only kind of eyeball it. The slope of this line gives the instantaneous velocity. The most useful role of this line is that students can tell when the velocity is increasing, decreasing, positive, negative, and nada.

[AL] You could discover the instantaneous velocity at each point along the graph and if you graphed each of those points, you would accept a graph of the velocity.

Practise Problems

16 .

Calculate the average velocity of the object shown in the graph below over the whole time interval.

1. 0.25 m/s
2. 0.31 m/south
3. 3.2 m/s
4. 4.00 m/s

17 .

True or Simulated: By taking the slope of the curve in the graph yous tin can verify that the velocity of the jet car is 125\,\text{grand/southward} at t = 20\,\text{southward}.

1. True

2. Imitation

Bank check Your Agreement

18 .

Which of the following information most movement can be determined by looking at a position vs. time graph that is a direct line?

1. frame of reference
2. average dispatch
3. velocity
4. direction of force practical

nineteen .

True or False: The position vs time graph of an object that is speeding upwards is a direct line.

1. Truthful

2. False

Instructor Back up

Instructor Back up

Utilize the Cheque Your Understanding questions to appraise students' accomplishment of the section'due south learning objectives. If students are struggling with a specific objective, the Bank check Your Understanding volition aid identify direct students to the relevant content.

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Source: https://openstax.org/books/physics/pages/2-3-position-vs-time-graphs

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