How To Find Change In Moment From A Position Time Graph

Section Learning Objectives

By the end of this section, you volition be able to do the following:

• Explain the meaning of slope and surface area in velocity vs. time graphs
• Solve problems using velocity vs. time graphs

Teacher Support

Teacher Support

The learning objectives in this section will aid your students master the following standards:

• (four) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The pupil is expected to:
  • (A) generate and interpret graphs and charts describing dissimilar types of motion, including the use of real-time technology such as motion detectors or photogates.

Section Key Terms

Teacher Support

Teacher Support

Ask students to use their knowledge of position graphs to construct velocity vs. time graphs. Alternatively, provide an example of a velocity vs. fourth dimension graph and enquire students what information can be derived from the graph. Ask—Is it the same information as in a position vs. time graph? How is the data portrayed differently? Is there any new data in a velocity vs. time graph?

Graphing Velocity every bit a Function of Time

Earlier, nosotros examined graphs of position versus time. Now, nosotros are going to build on that information as we look at graphs of velocity vs. fourth dimension. Velocity is the rate of change of displacement. Dispatch is the rate of change of velocity; nosotros will talk over acceleration more in some other chapter. These concepts are all very interrelated.

Virtual Physics

Maze Game

In this simulation yous volition use a vector diagram to manipulate a ball into a certain location without hitting a wall. Yous can manipulate the ball directly with position or by changing its velocity. Explore how these factors change the motion. If you would similar, you tin can put it on the a setting, as well. This is acceleration, which measures the rate of change of velocity. We will explore acceleration in more particular subsequently, merely it might be interesting to take a look at information technology here.

If a person takes 3 steps and ends up in the exact same place as their starting indicate, what must be truthful?

1. The three steps must have equal displacement

2. The displacement of the 3rd step is larger than the displacement of the first 2.

3. The average velocity must add up to zero.

4. The altitude and average velocity must add upwardly to cypher.

What tin can we acquire nigh motion by looking at velocity vs. time graphs? Let'southward render to our bulldoze to school, and await at a graph of position versus fourth dimension as shown in Figure 2.15.

Figure 2.15 A graph of position versus time for the drive to and from school is shown.

We assumed for our original calculation that your parent collection with a constant velocity to and from school. We now know that the car could not have gone from rest to a constant velocity without speeding upwards. And so the actual graph would be curved on either end, but permit's brand the same approximation as we did and so, anyway.

Tips For Success

Information technology is common in physics, specially at the early on learning stages, for sure things to be neglected, as we see here. This is considering information technology makes the concept clearer or the adding easier. Practicing physicists use these kinds of short-cuts, as well. It works out because commonly the matter being neglected is pocket-sized enough that it does not significantly affect the answer. In the earlier example, the amount of time it takes the car to speed up and reach its cruising velocity is very minor compared to the full time traveled.

Looking at this graph, and given what we learned, we can see that there are ii singled-out periods to the car's motion—the mode to school and the way back. The average velocity for the drive to school is 0.5 km/minute. We tin encounter that the boilerplate velocity for the drive dorsum is –0.v km/minute. If we plot the data showing velocity versus time, we become another graph (Effigy 2.16):

Figure 2.sixteen Graph of velocity versus fourth dimension for the drive to and from school.

We can learn a few things. First, nosotros tin derive a v versus t graph from a d versus t graph. 2d, if we have a straight-line position–time graph that is positively or negatively sloped, it will yield a horizontal velocity graph. In that location are a few other interesting things to note. Just as we could use a position vs. time graph to decide velocity, we can employ a velocity vs. fourth dimension graph to determine position. We know that five = d/t. If we apply a trivial algebra to re-conform the equation, nosotros see that d = v $\times$ t. In Effigy 2.16, nosotros take velocity on the y-centrality and fourth dimension along the x-axis. Let's accept just the first half of the move. Nosotros get 0.five km/minute $\times$ 10 minutes. The units for minutes cancel each other, and nosotros go 5 km, which is the deportation for the trip to schoolhouse. If we calculate the same for the return trip, we become –5 km. If nosotros add them together, nosotros run into that the net displacement for the whole trip is 0 km, which it should be considering we started and ended at the same identify.

Tips For Success

You can treat units just similar you treat numbers, then a km/km=i (or, nosotros say, it cancels out). This is proficient because it tin tell us whether or not we accept calculated everything with the correct units. For instance, if we cease up with m × s for velocity instead of m/s, we know that something has gone wrong, and nosotros need to check our math. This procedure is called dimensional analysis, and it is one of the best ways to check if your math makes sense in physics.

The expanse under a velocity bend represents the displacement. The velocity curve also tells united states whether the car is speeding upwards. In our before example, we stated that the velocity was abiding. Then, the motorcar is not speeding up. Graphically, y'all can see that the slope of these two lines is 0. This slope tells us that the car is non speeding up, or accelerating. We will do more with this information in a later chapter. For now, simply remember that the surface area nether the graph and the slope are the two important parts of the graph. Just similar nosotros could define a linear equation for the motion in a position vs. time graph, we can also ascertain 1 for a velocity vs. time graph. As we said, the gradient equals the acceleration, a. And in this graph, the y-intercept is v ₀. Thus, $v={\mathrm{five}}_0+at$ .

But what if the velocity is not constant? Let's expect back at our jet-car instance. At the beginning of the motility, as the car is speeding up, we saw that its position is a curve, as shown in Figure 2.17.

Figure ii.17 A graph is shown of the position of a jet-powered car during the fourth dimension span when information technology is speeding upward. The slope of a d vs. t graph is velocity. This is shown at two points. Instantaneous velocity at any indicate is the slope of the tangent at that point.

You lot do not have to do this, but you could, theoretically, take the instantaneous velocity at each signal on this graph. If you did, you would get Effigy ii.18, which is just a directly line with a positive slope.

Figure 2.18 The graph shows the velocity of a jet-powered motorcar during the time span when information technology is speeding up.

Again, if we take the slope of the velocity vs. time graph, we become the acceleration, the rate of change of the velocity. And, if we accept the surface area nether the slope, we get back to the displacement.

Teacher Support

Teacher Support

Teacher Demonstration

Render to the scenario of the drive to and from schoolhouse. Re-draw the V-shaped position graph. Ask the students what the velocity is at different times on that graph. Students should then exist able to see that the respective velocity graph is a horizontal line at 0.5km/minute and then a horizontal line at –0.5 km/minute. Then draw a few velocity graphs and see if they can go the corresponding position graph.

[OL] [AL] Accept students draw the relationship between the velocity and the position on these graphs. Ask—Can a velocity graph be used to discover the position? Can a velocity graph be used to notice anything else?

[AL] What is wrong with this graph? Inquire students whether the velocity could really be constant from rest or shift to negative so quickly. What would more realistic graphs look like? How inaccurate is it to ignore the not-abiding portion of the motion?

[OL] Students should exist able to see that if a position graph is a straight line, then the velocity graph will be a horizontal line. Likewise, the instantaneous velocity can be read off the velocity graph at any moment, merely more than steps are needed to calculate the average velocity.

[AL] Guide students in seeing that the area under the velocity curve is actually the position and the slope represents the rate of change of the velocity, but as the slope of the position line represents the rate of change of the position.

Solving Problems using Velocity–Time Graphs

About velocity vs. fourth dimension graphs will be directly lines. When this is the case, our calculations are fairly simple.

Worked Example

Using Velocity Graph to Calculate Some Stuff: Jet Car

Use this effigy to (a) detect the displacement of the jet car over the time shown (b) calculate the charge per unit of modify (acceleration) of the velocity. (c) give the instantaneous velocity at 5 s, and (d) calculate the average velocity over the interval shown.

Strategy

1. The displacement is given past finding the expanse under the line in the velocity vs. time graph.
2. The acceleration is given by finding the slope of the velocity graph.
3. The instantaneous velocity tin can just be read off of the graph.
4. To find the average velocity, recall that $v_{\text{avg}}=\frac{\Delta d}{\Delta t}=\frac{d_{\text{f}}-d_0}{t_{\text{f}}-t_0}$

Word

The average velocity we calculated hither makes sense if we look at the graph. 100m/s falls about halfway across the graph and since it is a direct line, we would expect about half the velocity to be above and half below.

Instructor Support

Teacher Support

The quantities solved for are slightly different in the different kinds of graphs, but students should begin to see that the process of analyzing or breaking down any of these graphs is similar. Ask—Where are the turning points in the motion? When is the object moving forward? What does a curve in the graph mean? Also, students should starting time to have an intuitive understanding of the relationship between position and velocity graphs.

Tips For Success

You can have negative position, velocity, and acceleration on a graph that describes the way the object is moving. You should never come across a graph with negative time on an axis. Why?

Most of the velocity vs. fourth dimension graphs we will expect at will be simple to interpret. Occasionally, we will look at curved graphs of velocity vs. time. More often, these curved graphs occur when something is speeding up, often from residuum. Let's look back at a more realistic velocity vs. time graph of the jet car's motion that takes this speeding up stage into account.

Figure 2.19 The graph shows a more accurate graph of the velocity of a jet-powered car during the time span when information technology is speeding up.

Worked Case

Using Curvy Velocity Graph to Calculate Some Stuff: jet car, Take 2

Employ Figure 2.nineteen to (a) detect the judge displacement of the jet car over the time shown, (b) calculate the instantaneous acceleration at t = 30 southward, (c) find the instantaneous velocity at 30 s, and (d) calculate the guess average velocity over the interval shown.

Strategy

1. Because this graph is an undefined curve, we take to approximate shapes over smaller intervals in order to find the areas.
2. Similar when nosotros were working with a curved displacement graph, we will demand to take a tangent line at the instant we are interested and apply that to calculate the instantaneous acceleration.
3. The instantaneous velocity tin still exist read off of the graph.
4. We volition find the average velocity the same way nosotros did in the previous example.

Discussion

This is a much more complicated process than the first trouble. If we were to use these estimates to come up with the average velocity over merely the first 30 southward nosotros would get about 191 1000/due south. By approximating that curve with a line, nosotros go an average velocity of 202.5 m/s. Depending on our purposes and how precise an answer we need, sometimes calling a curve a straight line is a worthwhile approximation.

Teacher Support

Teacher Support

Finding the tangent line can be a challenging concept for high school students, and they need to sympathize it theoretically. If you lot drew a regular bend inside of the bend at the point y'all are interested in, yous could describe a radius of that bend. The tangent line would be the line perpendicular to that radius.

[OL] Have the students compare this problem and the concluding one. Ask—What is the difference? When would y'all intendance nigh the more than accurate picture of the motion? And when would it actually not affair? Why would you lot ever want to look at a less authentic depiction of movement?

Practice Problems

20 .

Figure 2.20

Consider the velocity vs. time graph shown below of a person in an lift. Suppose the elevator is initially at rest. Information technology then speeds up for 3 seconds, maintains that velocity for 15 seconds, then slows down for 5 seconds until information technology stops. Detect the instantaneous velocity at t = 10 south and t = 23 south.

1. Instantaneous velocity at t = 10 south and t = 23 southward are 0 chiliad/s and 0 thou/s.
2. Instantaneous velocity at t = 10 s and t = 23 s are 0 chiliad/s and 3 m/s.
3. Instantaneous velocity at t = x south and t = 23 s are three m/due south and 0 yard/due south.
4. Instantaneous velocity at t = 10 s and t = 23 due south are 3 m/southward and 1.five chiliad/s.

21 .

Figure 2.21

Summate the net deportation and the average velocity of the elevator over the time interval shown.

1. Net displacement is 45 one thousand and boilerplate velocity is ii.10 m/s.
2. Cyberspace displacement is 45 thousand and average velocity is 2.28 m/due south.
3. Net displacement is 57 yard and average velocity is ii.66 yard/s.
4. Net displacement is 57 m and boilerplate velocity is 2.48 thou/south.

Snap Lab

Graphing Motion, Take Ii

In this activity, you will graph a moving ball'southward velocity vs. time.

• your graph from the before Graphing Motion Snap Lab!
• 1 piece of graph paper
• 1 pencil

Procedure

1. Take your graph from the earlier Graphing Motion Snap Lab! and use it to create a graph of velocity vs. fourth dimension.
2. Apply your graph to calculate the displacement.

22 .

Describe the graph and explain what it ways in terms of velocity and acceleration.

1. The graph shows a horizontal line indicating that the ball moved with a constant velocity, that is, it was non accelerating.

2. The graph shows a horizontal line indicating that the ball moved with a constant velocity, that is, it was accelerating.

3. The graph shows a horizontal line indicating that the ball moved with a variable velocity, that is, it was not accelerating.

4. The graph shows a horizontal line indicating that the ball moved with a variable velocity, that is, it was accelerating.

Instructor Support

Teacher Support

In this lab, students will use the displacement graph they drew in the last snap lab to create a velocity graph. If the rolling ball slowed down in the concluding snap lab, perhaps due to the ramp existence likewise depression, and then the graph may not show abiding velocity.

Check Your Understanding

23 .

What data could you lot obtain by looking at a velocity vs. time graph?

1. dispatch

2. direction of motion

3. reference frame of the movement

4. shortest path

24 .

How would yous use a position vs. time graph to construct a velocity vs. time graph and vice versa?

1. The slope of a position vs. time bend is used to construct a velocity vs. time curve, and the slope of a velocity vs. fourth dimension bend is used to construct a position vs. time curve.

2. The slope of a position vs. time bend is used to construct a velocity vs. fourth dimension curve, and the area of a velocity vs. fourth dimension curve is used to construct a position vs. time bend.

3. The area of a position vs. time curve is used to construct a velocity vs. time curve, and the slope of a velocity vs. time bend is used to construct a position vs. fourth dimension bend.

4. The surface area of a position vs. fourth dimension curve is used to construct a velocity vs. time curve, and the surface area of a velocity vs. time curve is used to construct a position vs. time curve.

Instructor Support

Teacher Support

Use the Cheque Your Understanding questions to assess students' achievement of the section's learning objectives. If students are struggling with a specific objective, the Check Your Understanding will aid direct students to the relevant content.

Source: https://openstax.org/books/physics/pages/2-4-velocity-vs-time-graphs

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