How To Find Change In Momentum From A Position Time Graph

Section Learning Objectives

By the end of this department, you will be able to do the following:

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

Instructor Support

Teacher Support

The learning objectives in this section will help your students chief the following standards:

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

Section Key Terms

dependent variable

independent variable

tangent

Teacher Support

Teacher Support

[BL] [OL] Depict a scenario, for example, in which yous launch a water rocket into the air. It goes up 150 ft, stops, then falls back to the earth. Have the students assess the situation. Where would they put their zero? What is the positive direction, and what is the negative direction? Have a student draw a film of the scenario on the board. So draw a position vs. fourth dimension graph describing the motion. Have students help you consummate the graph. Is the line direct? Is information technology curved? Does it modify direction? What can they tell by looking at the graph?

[AL] In one case the students accept looked at and analyzed the graph, see if they tin describe unlike scenarios in which the lines would exist direct instead of curved? Where the lines would be discontinuous?

Graphing Position equally a Function of Time

A graph, like a movie, is worth a yard words. Graphs not but comprise 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 have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against each other, the horizontal axis is usually considered the independent variable, and the vertical axis is the dependent variable. In algebra, yous would have referred to the horizontal axis equally the x-axis and the vertical centrality equally the y-axis. Equally in Effigy 2.10, a direct-line graph has the general form $y=g\mathrm{10}+b$ .

Here yard is the gradient, defined every bit the rising divided by the run (as seen in the figure) 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 situation in the existent earth, these quantities will accept on a specific significance, every bit nosotros will see below. (Figure 2.10.)

Effigy 2.10 The diagram shows a direct-line graph. The equation for the straight line is y equals mx + b.

In physics, fourth dimension is usually the contained variable. Other quantities, such as deportation, are said to depend upon information technology. A graph of position versus time, therefore, would accept position on the vertical centrality (dependent variable) and time on the horizontal axis (independent variable). In this case, to what would the slope and y-intercept refer? Permit'south await back at our original instance when studying distance and displacement.

The bulldoze to schoolhouse was 5 km from domicile. Allow's assume it took ten 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 wait like that shown in Figure two.11.

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

Every bit we said before, d ₀ = 0 because we telephone call dwelling our O and get-go calculating from there. In Figure 2.11, the line starts at d = 0, every bit well. This is the b in our equation for a directly line. Our initial position in a position versus fourth dimension graph is always the place where the graph crosses the ten-axis at t = 0. What is the gradient? The rise is the change in position, (i.e., displacement) and the run is the change in time. This relationship can also be written

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

Tips For Success

Sometimes, equally is the instance where nosotros graph both the trip to school and the render trip, the behavior of the graph looks different during different time intervals. If the graph looks like a series of direct lines, then you lot tin can calculate the boilerplate velocity for each time interval by looking at the gradient. If you lot and then want to summate the average velocity for the entire trip, you tin practice a weighted average.

Allow's look at another example. Effigy 2.12 shows a graph of position versus fourth dimension for a jet-powered automobile on a very flat dry lake bed in Nevada.

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

Using the relationship betwixt dependent and contained variables, we see that the slope in the graph in Effigy 2.12 is boilerplate velocity, v _avg and the intercept is displacement at time goose egg—that is, d ₀. Substituting these symbols into y = mx + b gives

or

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

ii.6

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

Teacher Support

Teacher Back up

Teacher Demonstration

Assist students learn what different graphs of displacement vs. fourth dimension look like.

[Visual] Ready a meter stick.

1. If you tin can find a remote control car, have ane student tape times equally yous transport the car frontwards forth the stick, and then backwards, then forward again with a abiding velocity.
2. Take the recorded times and the change in position and put them together.
3. Get the students to motorbus you 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 part where it is going backwards would have a negative slope.

[OL] Ask if the place that they take equally naught affects the graph.

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

[All] Discuss what can be uncovered from this graph. Students should be able to read the net deportation, just they can also employ the graph to determine 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 mensurate 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 larn virtually in the next chapter.

Snap Lab

Graphing Motion

In this action, you will release a ball down a ramp and graph the ball's displacement vs. fourth dimension.

• Choose an open location with lots of space to spread out and so at that place is less chance for tripping or falling due to rolling balls.

• ane ball
• i lath
• two or iii books
• one stopwatch
• 1 tape measure out
• 6 pieces of masking record
• 1 piece of graph paper
• 1 pencil

Procedure

1. Build a ramp by placing 1 end of the board on top of the stack of books. Adjust location, as necessary, until at that place is no obstacle along the direct line path from the bottom of the ramp until at least the next 3 g.
2. Mark distances of 0.5 thousand, ane.0 m, i.5 m, 2.0 g, 2.5 m, and 3.0 thou from the bottom of the ramp. Write the distances on the record.
3. Have one person have the role of the experimenter. This person will release the brawl from the top of the ramp. If the ball does not reach the 3.0 g marking, so increase the incline of the ramp by adding another book. Echo this Step as necessary.
4. Have the experimenter release the ball. Have a second person, the timer, begin timing the trial one time the ball reaches the lesser of the ramp and stop the timing in one case the brawl reaches 0.5 thou. Have a 3rd person, the recorder, record the time in a data table.
5. Echo Step 4, stopping the times at the distances of one.0 m, 1.v thou, 2.0 m, 2.five m, and 3.0 m from the bottom of the ramp.
6. Use your measurements of time and the deportation to make a position vs. time graph of the brawl's motion.
7. Echo Steps 4 through 6, with different people taking on the roles of experimenter, timer, and recorder. Do y'all go the same measurement values regardless of who releases the ball, measures the time, or records the outcome? Discuss possible causes of discrepancies, if whatever.

True or Imitation: The average speed of the brawl volition exist less than the average velocity of the ball.

1. Truthful

2. Faux

Teacher Support

Teacher Support

[BL] [OL] Emphasize that the motion in this lab is the motion of the brawl as information technology rolls along the floor. Enquire students where there zilch should be.

[AL] Inquire students what the graph would expect like if they began timing at the top versus the bottom of the ramp. Why would the graph look unlike? What might account for the departure?

[BL] [OL] Have the students compare the graphs made with different individuals taking on different roles. Enquire them to determine 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 between the ii sets of measurements with different individuals in each role?

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

Solving Issues Using Position vs. Time Graphs

And so how do we use graphs to solve for things we desire to know like velocity?

Worked Example

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

Observe the average velocity of the car whose position is graphed in Figure one.13.

Strategy

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

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

two.vii

Since the slope is constant here, whatever two points on the graph tin exist used to discover the slope.

Discussion

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

Instructor Support

Teacher Back up

If the graph of position is a straight line, then the just thing students need to know to summate the average velocity is the slope of the line, rise/run. They tin use whichever points on the line are most convenient.

But 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 backward? Can we figure out anything about its velocity from a graph of that kind of motion? Permit'southward have another wait at the jet-powered auto. The graph in Figure 2.xiii shows its move as it is getting upwards to speed after starting at rest. Fourth dimension starts at zero for this move (every bit if measured with a stopwatch), and the displacement and velocity are initially 200 grand and fifteen m/s, respectively.

Figure 2.thirteen The diagram shows a graph of the position of a jet-powered motorcar during the fourth dimension span when information technology is speeding upwardly. The gradient of a distance versus time graph is velocity. This is shown at ii points. Instantaneous velocity at any point is the slope of the tangent at that signal.

Effigy two.14 A U.S. Air Force jet car speeds down a track. (Matt Trostle, Flickr)

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

Worked Example

Using Position–Time Graph to Summate Average Velocity: Jet Machine, Take Two

Calculate the instantaneous velocity of the jet motorcar at a time of 25 s by finding the slope of the tangent line at point Q in Figure two.thirteen.

Strategy

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

Discussion

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

Teacher Back up

Teacher Support

A curved line is a more complicated case. Ascertain tangent as a line that touches a curve at only one point. Bear witness that as a directly line changes its bending adjacent to a bend, it actually hits the curve multiple times at the base, simply only one line will never touch at all. This line forms a correct angle to the radius of curvature, but at this level, they tin only kind of eyeball it. The slope of this line gives the instantaneous velocity. The near useful office of this line is that students tin tell when the velocity is increasing, decreasing, positive, negative, and zero.

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

Practice Problems

16 .

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

1. 0.25 m/southward
2. 0.31 chiliad/due south
3. iii.2 grand/s
4. 4.00 1000/s

17 .

True or Imitation: By taking the slope of the curve in the graph you can verify that the velocity of the jet car is 125\,\text{thou/south} at t = xx\,\text{s}.

1. Truthful

2. Faux

Check Your Understanding

18 .

Which of the following information well-nigh movement tin can be determined by looking at a position vs. fourth dimension graph that is a directly line?

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

xix .

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

1. Truthful

2. Faux

Teacher Support

Instructor Support

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

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

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