Home  ›  what information is provided by the slope of the two graphs to be obtained from the data

what information is provided by the slope of the two graphs to be obtained from the data

Written By Thursday, April 7, 2022 Add Comment Edit

Section Learning Objectives

By the end of this section, y'all will be able to do the following:

  • Explicate the meaning of slope in position vs. fourth dimension graphs
  • Solve problems using position vs. time graphs

Teacher Support

Teacher Support

The learning objectives in this department will help your students principal the following standards:

  • (4) Scientific discipline concepts. The student knows and applies the laws governing movement in a variety of situations. The student is expected to:
    • (A) generate and interpret graphs and charts describing unlike types of movement, including the use of real-fourth dimension technology such as move detectors or photogates.

Section Cardinal Terms

dependent variable independent variable tangent

Instructor Support

Teacher Back up

[BL] [OL] Describe a scenario, for case, in which you launch a water rocket into the air. Information technology goes up 150 ft, stops, and and then falls dorsum to the earth. Have the students assess the situation. Where would they put their zippo? What is the positive direction, and what is the negative direction? Have a student describe a picture of the scenario on the board. Then draw a position vs. fourth dimension graph describing the motion. Take students help you consummate the graph. Is the line directly? Is it curved? Does it change direction? What can they tell by looking at the graph?

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

Graphing Position as a Function of Time

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

Graphs in this text have perpendicular axes, one horizontal and the other vertical. When ii physical quantities are plotted against each other, the horizontal centrality is usually considered the independent 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 equally the y-axis. Equally in Figure 2.10, a straight-line graph has the general class y = m x + b y = m 10 + b .

Here m is the slope, defined as the rise 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-centrality. In terms of a physical situation in the real globe, these quantities will have on a specific significance, as we will see below. (Figure two.x.)

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

In physics, fourth dimension is normally the independent variable. Other quantities, such equally deportation, are said to depend upon information technology. A graph of position versus time, therefore, would have position on the vertical axis (dependent variable) and time on the horizontal axis (contained variable). In this case, to what would the slope and y-intercept refer? Let's wait back at our original example when studying distance and displacement.

The bulldoze to school was 5 km from home. Let's assume 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 time graph for this section of the trip would look similar that shown in Figure 2.eleven.

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.

Figure 2.xi A graph of position versus time for the drive to schoolhouse is shown. What would the graph look similar if we added the render trip?

As we said before, d 0 = 0 because nosotros telephone call home our O and offset calculating from there. In Effigy 2.eleven, the line starts at d = 0, as well. This is the b in our equation for a directly line. Our initial position in a position versus time graph is always the place where the graph crosses the ten-axis at t = 0. What is the slope? The rise is the modify in position, (i.eastward., displacement) and the run is the change in time. This relationship can also be written

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

Tips For Success

Sometimes, every bit is the case where we graph both the trip to schoolhouse and the return trip, the behavior of the graph looks unlike during different time intervals. If the graph looks like a series of directly lines, so you tin calculate the average velocity for each time interval by looking at the gradient. If you and so want to summate the boilerplate velocity for the entire trip, yous can do a weighted average.

Permit's look at some other example. Figure 2.12 shows a graph of position versus time for a jet-powered car on a very flat dry 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, we see that the slope in the graph in Figure 2.12 is average velocity, v avg and the intercept is displacement at time zero—that is, d 0. Substituting these symbols into y = mx + b gives

or

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

2.6

Thus a graph of position versus fourth dimension gives a general relationship amidst displacement, velocity, and time, as well equally giving detailed numerical information about a specific situation. From the figure we can see that the car has a position of 400 m at t = 0 s, 650 k at t = 1.0 s, and so on. And we tin can learn virtually the object's velocity, as well.

Instructor Support

Instructor Support

Instructor Demonstration

Aid students learn what different graphs of displacement vs. fourth dimension expect like.

[Visual] Ready a meter stick.

  1. If you can find a remote control machine, have i student record times equally yous send the car forward along the stick, then backwards, then forward once again with a constant velocity.
  2. Take the recorded times and the modify in position and put them together.
  3. Go the students to passenger vehicle you to draw a position vs. time graph.

Each leg of the journey should exist a direct line with a different slope. The parts where the auto was going forwards should have a positive slope. The part where it is going backwards would have a negative slope.

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

[AL] Is it realistic to draw whatsoever position graph that starts at rest without some bend 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, but they can as well employ the graph to decide the full distance traveled. Then 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 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 dispatch which they volition acquire nearly in the next affiliate.

Snap Lab

Graphing Motion

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

  • Choose an open location with lots of space to spread out so in that location is less take a chance for tripping or falling due to rolling balls.
  • 1 brawl
  • 1 board
  • two or 3 books
  • 1 stopwatch
  • 1 tape measure out
  • half dozen pieces of masking tape
  • 1 piece of graph paper
  • 1 pencil

Procedure

  1. Build a ramp by placing one end of the board on superlative of the stack of books. Adjust location, as necessary, until there is no obstacle along the straight line path from the bottom of the ramp until at least the next three thousand.
  2. Mark distances of 0.5 m, 1.0 chiliad, 1.5 m, 2.0 m, 2.v m, and 3.0 m from the lesser of the ramp. Write the distances on the tape.
  3. Have 1 person take the role of the experimenter. This person will release the ball from the top of the ramp. If the ball does not achieve the iii.0 m marking, then increase the incline of the ramp past adding some other book. Echo this Step as necessary.
  4. Have the experimenter release the ball. Have a second person, the timer, brainstorm timing the trial in one case the ball reaches the bottom of the ramp and stop the timing once the ball reaches 0.5 m. Take a third person, the recorder, tape the time in a data table.
  5. Echo Stride 4, stopping the times at the distances of 1.0 one thousand, 1.v m, 2.0 m, two.five grand, and 3.0 chiliad from the bottom of the ramp.
  6. Apply your measurements of time and the deportation to make a position vs. fourth dimension graph of the brawl's movement.
  7. Echo Steps 4 through 6, with different people taking on the roles of experimenter, timer, and recorder. Do you get the aforementioned measurement values regardless of who releases the ball, measures the fourth dimension, or records the result? Discuss possible causes of discrepancies, if any.

True or False: The boilerplate speed of the ball volition be less than the boilerplate velocity of the ball.

  1. True

  2. Faux

Instructor Support

Instructor Support

[BL] [OL] Emphasize that the movement in this lab is the move of the ball every bit information technology rolls along the floor. Inquire students where there zero should be.

[AL] Ask students what the graph would wait like if they began timing at the pinnacle versus the lesser of the ramp. Why would the graph look dissimilar? What might account for the deviation?

[BL] [OL] Accept the students compare the graphs made with unlike individuals taking on different roles. Ask them to make up one's mind 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 at that place systematic differences? Why might there be systematic differences betwixt the two sets of measurements with different individuals in each role?

[BL] [OL] Accept the students compare the graphs fabricated 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? Exercise the differences announced to be random, or are in that location systematic differences? Why might in that location be systematic differences between the ii sets of measurements with different individuals in each part?

Solving Problems Using Position vs. Time Graphs

So how do we use graphs to solve for things we want to know like velocity?

Worked Example

Using Position–Fourth dimension Graph to Calculate Average Velocity: Jet Automobile

Find the average velocity of the car whose position is graphed in Figure 1.13.

Strategy

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

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

2.7

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

Discussion

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

Teacher Back up

Teacher Support

If the graph of position is a straight line, then the merely matter students need to know to summate the boilerplate velocity is the slope of the line, rise/run. They tin can utilize whichever points on the line are most user-friendly.

But what if the graph of the position is more than complicated than a straight line? What if the object speeds up or turns around and goes astern? Can we figure out anything almost its velocity from a graph of that kind of motility? Let's accept another await at the jet-powered motorcar. The graph in Figure two.13 shows its motion as it is getting up to speed after starting at residue. Time starts at zero for this motion (equally if measured with a stopwatch), and the displacement and velocity are initially 200 m and fifteen m/s, 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.thirteen The diagram shows a graph of the position of a jet-powered car during the time span when information technology is speeding upward. The slope of a distance versus fourth dimension graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that point.

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

Effigy two.14 A U.Due south. Air Force jet automobile speeds downwardly a track. (Matt Trostle, Flickr)

The graph of position versus fourth dimension in Figure 2.13 is a curve rather than a direct line. The gradient of the bend becomes steeper as fourth dimension progresses, showing that the velocity is increasing over time. The slope at whatsoever signal on a position-versus-time graph is the instantaneous velocity at that point. It is found by drawing a directly line tangent to the bend at the point of interest and taking the slope of this directly line. Tangent lines are shown for ii points in Figure ii.13. The boilerplate velocity is the cyberspace displacement divided by the time traveled.

Worked Example

Using Position–Time Graph to Calculate Boilerplate Velocity: Jet Car, Have 2

Summate the instantaneous velocity of the jet car at a time of 25 s by finding the gradient of the tangent line at signal Q in Figure 2.13.

Strategy

The slope of a curve at a indicate is equal to the slope of a straight line tangent to the curve at that signal.

Word

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

Instructor Support

Teacher Back up

A curved line is a more complicated instance. Define tangent as a line that touches a curve at merely one indicate. Evidence that as a straight line changes its angle next to a curve, it really hits the curve multiple times at the base, but only one line will never touch at all. This line forms a correct angle 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 can tell when the velocity is increasing, decreasing, positive, negative, and zero.

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

Practice Problems

xvi .

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

no alt text

  1. 0.25 one thousand/s
  2. 0.31 m/south
  3. three.2 1000/s
  4. four.00 m/south

17 .

True or Faux: By taking the gradient of the curve in the graph you can verify that the velocity of the jet car is 125\,\text{m/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

Cheque Your Agreement

18 .

Which of the following information about motility tin exist determined by looking at a position vs. fourth dimension graph that is a straight line?

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

19 .

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

  1. True

  2. False

Teacher Support

Teacher Back up

Utilize the Check Your Understanding questions to appraise students' achievement of the section's learning objectives. If students are struggling with a specific objective, the Check Your Agreement will help identify directly students to the relevant content.

ortizvirstal.blogspot.com

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

0 Response to "what information is provided by the slope of the two graphs to be obtained from the data"

Post a Comment

Subscribe to: Post Comments (Atom)

Iklan Atas Artikel

Iklan Tengah Artikel 1

Iklan Tengah Artikel 2

Iklan Bawah Artikel