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Shows a Motion Diagram With the Clock Reading Shown at Each Position

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Department Learning Objectives

Past the finish of this section, y'all volition be able to do the following:

  • Explain the significant of slope in position vs. time graphs
  • Solve problems using position vs. time graphs

Teacher Support

Teacher Support

The learning objectives in this department volition help your students master the post-obit standards:

  • (4) Science concepts. The student knows and applies the laws governing move in a variety of situations. The pupil is expected to:
    • (A) generate and translate graphs and charts describing different types of motility, including the use of real-time engineering science such as movement detectors or photogates.

Section Key Terms

dependent variable independent variable tangent

Teacher Support

Teacher Support

[BL] [OL] Draw a scenario, for example, in which yous launch a h2o rocket into the air. Information technology goes upward 150 ft, stops, and and so falls dorsum to the globe. Have the students assess the situation. Where would they put their zero? What is the positive management, and what is the negative direction? Have a pupil draw a picture of the scenario on the board. Then draw a position vs. time graph describing the motility. Have students aid yous consummate the graph. Is the line straight? Is it curved? Does information technology change direction? What can they tell past looking at the graph?

[AL] Once the students have looked at and analyzed the graph, see if they can draw different scenarios in which the lines would be straight instead of curved? Where the lines would exist discontinuous?

Graphing Position as a Function of Time

A graph, like a motion-picture show, is worth a thousand words. Graphs not just contain numerical information, they also reveal relationships betwixt physical quantities. In this section, we will investigate kinematics by analyzing graphs of position over time.

Graphs in this text accept perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted confronting each other, the horizontal centrality is normally considered the contained variable, and the vertical axis is the dependent variable. In algebra, you would have referred to the horizontal axis as the ten-axis and the vertical axis as the y-axis. As in Effigy two.ten, a straight-line graph has the general course y = m x + b y = m x + b .

Here m is the gradient, defined as the rising divided past the run (as seen in the effigy) of the straight line. The letter b is the y-intercept which is the point at which the line crosses the vertical, y-axis. In terms of a physical state of affairs in the real world, these quantities will take on a specific significance, equally we volition run into below. (Figure 2.10.)

A line graph is shown. The plotted line has a positive slope. The point of intersection with the y-axis is labeled intercept, b. The distance between the graphed line and the x-axis at the intercept is also labeled b. The line is labeled with the equation y equals m x plus b. The equation slope equals rise over run equals change in x over change in y equals m is also shown.

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

In physics, time is ordinarily the contained variable. Other quantities, such every bit 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 centrality (contained variable). In this case, to what would the gradient and y-intercept refer? Let's await dorsum at our original example when studying altitude and displacement.

The drive to school was 5 km from home. Allow's presume it took 10 minutes to brand the drive and that your parent was driving at a abiding velocity the whole fourth dimension. The position versus time graph for this section of the trip would look like that shown in Figure two.11.

A line graph of position versus time is shown. The scale on the x-axis is from zero to twelve in increments of one, and is labeled time in minutes. The scale on the y-axis is from zero to twelve in increments of one and is labeled position in kilometers. A line of the equation y equals point five x is graphed and ends at ten minutes.

Effigy 2.11 A graph of position versus fourth dimension for the drive to school is shown. What would the graph look like if nosotros added the render trip?

As nosotros said before, d 0 = 0 because we telephone call dwelling house our O and start computing from there. In Figure 2.xi, the line starts at d = 0, as well. This is the b in our equation for a direct line. Our initial position in a position versus time graph is always the place where the graph crosses the x-axis at t = 0. What is the slope? The rise is the change in position, (i.eastward., displacement) and the run is the change in time. This relationship can also exist written

This relationship was how we defined boilerplate velocity. Therefore, the slope in a d versus t graph, is the boilerplate velocity.

Tips For Success

Sometimes, as is the example where we graph both the trip to schoolhouse and the render trip, the behavior of the graph looks different during unlike fourth dimension intervals. If the graph looks like a series of directly lines, so you tin calculate the boilerplate velocity for each time interval by looking at the slope. If you then want to calculate the average velocity for the unabridged trip, you can practice a weighted boilerplate.

Let's await at some other instance. Figure two.12 shows a graph of position versus time for a jet-powered auto on a very flat dry out lake bed in Nevada.

A line graph is shown. The x-axis is labeled time in seconds and has a scale from zero to eight in increments of one. The y-axis is labeled position, d, in meters and has a scale from zero to two thousand four hundred in increments of four hundred. A line is plotted that intersects the following data points: zero, four hundred (labeled d 0 equals four hundred), point five, five hundred twenty-five; and six point four, two thousand (labeled y equals d o plus bar over v times t). The line is labeled Slope equals bar over v equals change in d over change in t. The distance between points point five, five hundred twenty-five and six point four, two thousand is labeled change in t. The distance between points six point four, five hundred twenty-five and six point four, two thousand is labeled change in d.

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

Using the relationship between dependent and independent variables, nosotros see that the slope in the graph in Figure ii.12 is boilerplate velocity, five avg and the intercept is deportation at fourth dimension zero—that is, d 0. Substituting these symbols into y = mx + b gives

or

d = d 0 + v t . d = d 0 + five t .

2.6

Thus a graph of position versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information well-nigh a specific situation. From the effigy we can see that the car has a position of 400 g at t = 0 s, 650 m at t = 1.0 s, and and then on. And we can learn about the object'south velocity, as well.

Instructor Support

Teacher Support

Teacher Sit-in

Help students learn what different graphs of displacement vs. time look like.

[Visual] Gear up up a meter stick.

  1. If y'all tin can find a remote command machine, take one student record times as y'all send the machine forward along the stick, and then backwards, so forrard over again with a constant velocity.
  2. Take the recorded times and the change in position and put them together.
  3. Get the students to coach yous to depict a position vs. time graph.

Each leg of the journey should be a straight line with a different slope. The parts where the auto was going forrard should have a positive slope. The part where it is going backwards would take a negative gradient.

[OL] Inquire if the place that they take as zilch affects the graph.

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

[All] Discuss what can be uncovered from this graph. Students should be able to read the net displacement, just they tin can also use the graph to determine the total distance traveled. Then inquire how the speed or velocity is reflected in this graph. Direct students in seeing that the steepness of the line (gradient) is a measure of the speed and that the management of the slope is the direction of the motion.

[AL] Some students might recognize that a curve in the line represents a sort of slope of the slope, a preview of acceleration which they will learn about in the adjacent affiliate.

Snap Lab

Graphing Motion

In this activeness, you will release a ball down a ramp and graph the ball'due south displacement vs. time.

  • Choose an open location with lots of space to spread out so there is less take a chance for tripping or falling due to rolling balls.
  • ane ball
  • 1 lath
  • two or 3 books
  • 1 stopwatch
  • 1 tape measure
  • 6 pieces of masking tape
  • 1 piece of graph paper
  • 1 pencil

Process

  1. Build a ramp by placing one end of the board on top of the stack of books. Adjust location, as necessary, until there is no obstruction along the straight line path from the bottom of the ramp until at least the side by side 3 m.
  2. Marker distances of 0.5 m, i.0 m, one.five m, 2.0 m, 2.v m, and 3.0 m from the bottom of the ramp. Write the distances on the tape.
  3. Have i person take the role of the experimenter. This person will release the brawl from the meridian of the ramp. If the ball does not attain the 3.0 chiliad mark, then increase the incline of the ramp by adding another book. Repeat this Step equally necessary.
  4. Have 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 0.5 m. Have a tertiary person, the recorder, record the time in a data tabular array.
  5. Repeat Stride 4, stopping the times at the distances of ane.0 thou, 1.v 1000, 2.0 thousand, 2.5 k, and 3.0 m from the bottom of the ramp.
  6. Use your measurements of fourth dimension and the displacement to make a position vs. time graph of the brawl's movement.
  7. Repeat Steps four through 6, with dissimilar people taking on the roles of experimenter, timer, and recorder. Practise you lot get the same measurement values regardless of who releases the brawl, measures the time, or records the event? Talk over possible causes of discrepancies, if whatever.

True or Fake: The average speed of the ball will be less than the boilerplate velocity of the ball.

  1. True

  2. False

Teacher Support

Teacher Back up

[BL] [OL] Emphasize that the motility in this lab is the motion of the ball as it rolls along the flooring. Ask students where in that location aught should exist.

[AL] Ask students what the graph would look like if they began timing at the top versus the lesser of the ramp. Why would the graph look different? What might account for the difference?

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

[BL] [OL] Have the students compare the graphs made with unlike individuals taking on dissimilar roles. Enquire them to determine and compare average speeds for each interval. What were the absolute differences in speeds, and what were the pct 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 unlike individuals in each role?

Solving Problems Using Position vs. Fourth dimension Graphs

And so how do we utilise graphs to solve for things nosotros want to know similar velocity?

Worked Case

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

Discover the average velocity of the automobile whose position is graphed in Effigy 1.13.

Strategy

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

slope = Δ d Δ t = v slope = Δ d Δ t = five

2.7

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

Discussion

This is an impressively high country speed (900 km/h, or almost 560 mi/h): much greater than the typical highway speed limit of 27 m/s or 96 km/h, but considerably shy of the record of 343 one thousand/s or i,234 km/h, prepare in 1997.

Teacher Back up

Teacher Support

If the graph of position is a straight line, then the but thing students demand to know to calculate the average velocity is the gradient of the line, ascension/run. They can use whichever points on the line are virtually convenient.

Only what if the graph of the position is more complicated than a straight line? What if the object speeds up or turns effectually and goes astern? Can nosotros figure out anything virtually its velocity from a graph of that kind of motion? Let's have another wait at the jet-powered car. The graph in Figure 2.13 shows its movement as information technology is getting upwards to speed after starting at rest. Time starts at nothing for this motion (as if measured with a stopwatch), and the deportation and velocity are initially 200 1000 and 15 m/southward, respectively.

A line graph titled Jet Car Displacement is shown. The x-axis is labeled time, t, in seconds and has a scale from zero to forty on increments of ten. The y-axis is labeled displacement, x, in meters and has a scale from zero to three thousand five hundred in increments of five hundred. The following approximate data points are plotted, resulting in a line that curves upward: eight, two hundred fifty; ten, five hundred; fifteen, one thousand; twenty, one thousand five hundred; twenty-five, two thousand; thirty, three thousand. A right triangle is drawn at points eight, two hundred fifty; twelve, two hundred fifty, and twelve seven hundred fifty. The legs are labeled change in tp and change in dp. Point ten, five hundred is labeled P. Another right triangle is drawn at points twenty, one thousand five hundred; thirty, one thousand five hundred, and thirty, three thousand. The legs are labeled change in tq and change in dq. Point twenty-five, two thousand is labeled Q.

Figure 2.13 The diagram shows a graph of the position of a jet-powered automobile during the fourth dimension span when it is speeding up. The slope of a altitude versus fourth dimension graph is velocity. This is shown at ii points. Instantaneous velocity at any signal is the slope of the tangent at that betoken.

A photograph of a United States Air Force jet car is shown speeding down a track. Smoke is billowing from the back end.

Figure two.14 A U.South. Air Forcefulness jet car 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 gradient of the curve becomes steeper as fourth dimension progresses, showing that the velocity is increasing over fourth dimension. The slope at whatever point on a position-versus-time graph is the instantaneous velocity at that point. Information technology is establish past drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for ii points in Figure two.thirteen. The boilerplate velocity is the net displacement divided by the time traveled.

Worked Example

Using Position–Fourth dimension Graph to Calculate Average Velocity: Jet Car, Take 2

Calculate the instantaneous velocity of the jet car at a time of 25 southward past finding the gradient of the tangent line at point Q in Figure ii.xiii.

Strategy

The gradient of a curve at a point is equal to the slope of a straight line tangent to the curve at that point.

Give-and-take

The unabridged graph of v versus t tin be obtained in this mode.

Instructor Back up

Instructor Back up

A curved line is a more complicated instance. Define tangent every bit a line that touches a curve at merely one point. Prove that as a straight line changes its bending adjacent to a curve, information technology actually hits the bend multiple times at the base of operations, just only i line will never affect at all. This line forms a correct bending to the radius of curvature, but at this level, they can just kind of eyeball it. The slope of this line gives the instantaneous velocity. The most useful part of this line is that students tin can tell when the velocity is increasing, decreasing, positive, negative, and nothing.

[AL] Y'all could detect the instantaneous velocity at each point along the graph and if you graphed each of those points, you would take a graph of the velocity.

Do Bug

16 .

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

no alt text

  1. 0.25 1000/s
  2. 0.31 one thousand/s
  3. iii.ii g/southward
  4. 4.00 thou/southward

17 .

True or False: By taking the slope of the bend in the graph you can verify that the velocity of the jet car is 125\,\text{yard/s} at t = 20\,\text{s}.

A line graph titled Position versus Time is shown. The x-axis is labeled time in seconds with a scale from zero to thirty-five in increments of five. The y-axis is labeled position in meters with a scale from zero to three thousand five hundred in increments of five hundred. Plotted points include zero, two hundred fifty, twenty, one thousand five hundred, and thirty, three thousand. The graphed line curves upward.

  1. Truthful

  2. False

Check Your Agreement

18 .

Which of the following information about motion can be determined past looking at a position vs. time graph that is a straight line?

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

19 .

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

  1. True

  2. Faux

Teacher Support

Teacher Support

Use the Check Your Understanding questions to assess students' achievement of the section'southward learning objectives. If students are struggling with a specific objective, the Check Your Agreement will assist identify straight 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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