Explain the definition of slope and area in velocity vs. Time graphsSolve difficulties using velocity vs. Time graphs
Teacher Support

The discovering objectives in this ar will help your students grasp the complying with standards:

(4)Science concepts. The student knows and also applies the laws governing movement in a variety of situations. The college student is intended to: (A) generate and also interpret graphs and charts relenten different types of motion, including the use of real-time modern technology such as movement detectors or photogates.
Teacher Support

Ask college student to usage their expertise of position graphs to construct velocity vs. Time graphs. Alternatively, carry out an example of a velocity vs. Time graph and ask student what information can be obtained from the graph. Ask—Is the the same information as in a position vs. Time graph? just how is the information depicted differently? Is there any brand-new information in a velocity vs. Time graph?

Graphing Velocity together a role of Time

Earlier, us examined graphs of position versus time. Now, we are going to construct on that information as we look in ~ graphs of velocity vs. Time. Velocity is the rate of adjust of displacement. Acceleration is the price of readjust of velocity; we will comment on acceleration an ext in another chapter. These concepts are all an extremely interrelated.

You are watching: What does the area under a velocity vs time graph represent

Maze Game

In this simulation girlfriend will usage a vector diagram come manipulate a ball right into a specific location there is no hitting a wall. You have the right to manipulate the ball directly with place or by transforming its velocity. Discover how this factors readjust the motion. If you would certainly like, you have the right to put the on the a setting, as well. This is acceleration, which procedures the price of change of velocity. Us will explore acceleration in an ext detail later, but it can be exciting to take it a look in ~ it here.

If a person takes 3 steps and also ends up in the exact same place as their beginning point, what should be true?

What can we learn about motion by looking at velocity vs. Time graphs? Let’s return to our journey to school, and also look at a graph of place versus time as shown in figure 2.15.


We assumed for our initial calculation that your parent drove with a continuous velocity to and from school. We now recognize that the car could not have gone from remainder to a consistent velocity without speeding up. Therefore the really graph would be bent on either end, but let’s do the same approximation as we walk then, anyway.

It is common in physics, especially at the at an early stage learning stages, for particular things to be neglected, as we watch here. This is because it renders the concept clearer or the calculation easier. Practicing physicists use these type of short-cuts, as well. It works out since usually the thing being neglected is little enough that it does no significantly affect the answer. In the previously example, the lot of time that takes the auto to rate up and also reach its setravel velocity is very tiny compared come the full time traveled.

Looking at this graph, and also given what we learned, we can see the there are two distinct periods to the car’s motion—the way to school and also the means back. The mean velocity for the drive to college is 0.5 km/minute. We have the right to see that the mean velocity for the drive back is –0.5 km/minute. If us plot the data reflecting velocity matches time, we get an additional graph (Figure 2.16):


number 2.16 Graph that velocity versus time because that the journey to and also from school.

We can learn a few things. First, we have the right to derive a v versus t graph from a d versus t graph. Second, if we have actually a straight-line position–time graph that is positively or negative sloped, it will yield a horizontal velocity graph. There space a couple of other exciting things to note. Just as we could use a place vs. Time graph to identify velocity, we have the right to use a velocity vs. Time graph to identify position. We know that v = d/t. If we usage a little algebra to re-arrange the equation, we view that d = v ×× t. In number 2.16, we have velocity on the y-axis and also time along the x-axis. Stop take just the very first half that the motion. We get 0.5 km/minute ×× 10 minutes. The systems for minutes cancel every other, and also we gain 5 km, which is the displacement for the expedition to school. If us calculate the very same for the return trip, we gain –5 km. If we include them together, we watch that the network displacement for the totality trip is 0 km, which it have to be because we started and ended in ~ the very same place.

Tips for Success

You deserve to treat units as with you act numbers, for this reason a km/km=1 (or, us say, it cancels out). This is great because it deserve to tell united state whether or no we have actually calculated whatever with the correct units. For instance, if we end up v m × s because that velocity instead of m/s, we recognize that something has actually gone wrong, and also we require to check our math. This process is dubbed dimensional analysis, and it is among the best ways to check if your math renders sense in physics.

The area under a velocity curve to represent the displacement. The velocity curve likewise tells united state whether the car is speeding up. In our previously example, we stated that the velocity to be constant. So, the automobile is no speeding up. Graphically, you can see that the slope of these 2 lines is 0. This slope tells united state that the automobile is not speeding up, or accelerating. We will do much more with this info in a later chapter. For now, simply remember that the area under the graph and also the slope room the two vital parts the the graph. Just like we could define a straight equation for the motion in a position vs. Time graph, us can also define one because that a velocity vs. Time graph. Together we said, the slope amounts to the acceleration, a. And also in this graph, the y-intercept is v0. Thus, v= v 0 +at v= v 0 +at .

But what if the velocity is no constant? let’s look ago at our jet-car example. In ~ the beginning of the motion, as the car is speeding up, we observed that its position is a curve, as presented in figure 2.17.


figure 2.17 A graph is shown of the place of a jet-powered auto during the time span when the is speeding up. The steep of a d vs. T graph is velocity. This is presented at two points. Instantaneous velocity in ~ any allude is the slope of the tangent at the point.

You execute not need to do this, but you could, theoretically, take it the instantaneous velocity in ~ each point on this graph. If girlfriend did, you would get number 2.18, i beg your pardon is just a right line with a hopeful slope.


figure 2.18 The graph shows the velocity the a jet-powered automobile during the time span when the is speeding up.

Again, if us take the slope of the velocity vs. Time graph, we get the acceleration, the rate of adjust of the velocity. And, if we take the area under the slope, us get ago to the displacement.

Teacher Support

Teacher Support

Teacher Demonstration

Return to the scenario of the drive to and also from school. Re-draw the V-shaped position graph. Asking the student what the velocity is at different times on the graph. Students need to then have the ability to see the the equivalent velocity graph is a horizontal line at 0.5km/minute and then a horizontal heat at –0.5 km/minute. Then attract a couple of velocity graphs and see if lock can obtain the equivalent position graph.

    have actually students describe the relationship between the velocity and also the position on these graphs. Ask—Can a velocity graph be supplied to uncover the position? have the right to a velocity graph be offered to discover anything else?

    What is wrong v this graph? ask students whether the velocity can actually be consistent from rest or change to negative so quickly. What would much more realistic graphs look like? just how inaccurate is the to overlook the non-constant section of the motion?

      college student should have the ability to see the if a place graph is a right line, climate the velocity graph will be a horizontal line. Also, the instantaneous velocity have the right to be check out off the velocity graph at any type of moment, but an ext steps are required to calculation the mean velocity.

      overview students in seeing that the area under the velocity curve is actually the position and also the slope to represent the rate of readjust of the velocity, just as the slope of the position line to represent the rate of change of the position.

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      analyze the form of the area to it is in calculated. In this case, the area is made up of a rectangle in between 0 and also 20 m/s stretching to 30 s. The area the a rectangle is size ×× width. Therefore, the area that this piece is 600 m.Above that is a triangle whose basic is 30 s and height is 140 m/s. The area the a triangle is 0.5 ×× size ×× width. The area the this piece, therefore, is 2,100 m. Include them together to obtain a net displacement that 2,700 m. Take 2 points top top the velocity line. Say, t = 5 s and t = 25 s. In ~ t = 5 s, the worth of v = 40 m/s. In ~ t = 25 s, v = 140 m/s. Uncover the slope. a = Δv Δt = 100m/s 20s = 5 m/s 2 a = Δv Δt = 100m/s 20s = 5 m/s 2 The instantaneous velocity at t = 5 s, together we found in part (b) is just 40 m/s.Find the net displacement, i beg your pardon we uncovered in component (a) to be 2,700 m.Find the complete time which for this case is 30 s.Divide 2,700 m/30 s = 90 m/s.