### Learning Objectives

By the end of this section, friend will be able to do the following:

Explain the an interpretation of slope in position vs. Time graphsSolve troubles using position vs. Time graphsSection an essential Termsdependent variable | independent variable | tangent |

### Graphing place as a duty of Time

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

You are watching: What is the slope of a position vs time graph

Graphs in this text have actually perpendicular axes, one horizontal and the various other vertical. As soon as two physical quantities are plotted against each other, the horizontal axis is usually taken into consideration the live independence variable, and also the vertical axis is the dependency variable. In algebra, girlfriend would have referred come the horizontal axis together the *x*-axis and the upright axis as the *y*-axis. As in figure 2.11, a straight-line graph has actually the general form y=mx+by=mx+b.

Here *m* is the slope, defined as the rise divided by the run (as checked out in the figure) of the straight line. The letter *b* is the *y*-intercept i m sorry is the allude at which the line the cross the vertical, *y*-axis. In terms of a physical situation in the real world, these amounts will take it on a details significance, as we will check out below. (Figure 2.11.)

Figure 2.11 The diagram mirrors a straight-line graph. The equation for the right line is

*y*equates to

*mx*+

*b*.

In physics, time is generally the elevation variable. Various other quantities, such together displacement, are said to rely upon it. A graph of place versus time, therefore, would have position ~ above the vertical axis (dependent variable) and time top top the horizontal axis (independent variable). In this case, come what would certainly the slope and also *y*-intercept refer? let’s look ago at ours original instance when studying distance and also displacement.

The drive to college was 5 km from home. Stop assume it took 10 minutes to do the drive and also that your parent was driving at a continuous velocity the totality time. The position versus time graph for this ar of the expedition would look prefer that displayed in number 2.12.

Figure 2.12 A graph of place versus time because that the journey to school is shown. What would certainly the graph look favor if we added the return trip?

As we stated before, **d**0 = 0 since we contact home ours *O* and also start calculating native there. In number 2.12, the heat starts in ~ **d** = 0, as well. This is the *b* in ours equation because that a straight line. Our initial position in a place versus time graph is constantly the ar where the graph the cross the *x*-axis in ~ *t* = 0. What is the slope? The *rise* is the readjust in position, (i.e., displacement) and the *run* is the readjust in time. This relationship can additionally be written

This relationship was how we identified average velocity. Therefore, the steep in a **d** versus *t* graph, is the median velocity.

Sometimes, as is the instance where us graph both the pilgrimage to school and the return trip, the actions of the graph looks different during different time intervals. If the graph looks prefer a series of directly lines, then you have the right to calculate the mean velocity for each time interval by looking at the slope. If you then want to calculation the typical velocity because that the entire trip, you can do a weighted average.

Let’s watch at one more example. Number 2.13 mirrors a graph of position versus time for a jet-powered vehicle on a really flat dry lake bed in Nevada.

Figure 2.13 The diagram reflects a graph of place versus time for a jet-powered auto on the Bonneville Salt Flats.

Using the relationship in between dependent and also independent variables, we view that the slope in the graph in number 2.13 is average velocity, **v**avg and the intercept is displacement in ~ time zero—that is, **d**0. Substituting this symbols right into *y* = *mx* + *b* gives

2.5d=vt+d0d=vt+d0

or

2.6d=d0+vt.d=d0+vt.

Thus a graph of place versus time gives a basic relationship among displacement, velocity, and time, and giving detailed numerical information around a certain situation. Native the figure we have the right to see that the automobile has a position of 400 m at *t* = 0 s, 650 m in ~ *t* = 1.0 s, and also so on. And also we can learn about the object’s velocity, together well.

Graphing Motion

In this activity, you will release a ball down a ramp and also graph the ball’s displacement vs. Time.

Choose one open location with lots of space to spread out out so there is less chance because that tripping or falling as result of rolling balls.

1 ball1 board2 or 3 books1 stopwatch1 ice measure6 pieces of masking tape1 item of graph paper1 pencil

Build a ramp by put one end of the board on top of the stack of books. Adjust location, as necessary, until there is no obstacle follow me the directly line route from the bottom that the ramp till at least the next 3 m. Mark ranges of 0.5 m, 1.0 m, 1.5 m, 2.0 m, 2.5 m, and 3.0 m from the bottom of the ramp. Create the ranges on the tape.Have one person take the duty of the experimenter. This human will release the sphere from the peak of the ramp. If the round does no reach the 3.0 m mark, then increase the incline of the ramp by adding another book. Repeat this step as necessary.Have the experimenter relax the ball. Have actually a 2nd person, the timer, begin timing the trial once the sphere reaches the bottom of the ramp and stop the timing once the sphere reaches 0.5 m. Have actually a third person, the recorder, document the time in a data table.Repeat step 4, protecting against the times at the ranges of 1.0 m, 1.5 m, 2.0 m, 2.5 m, and also 3.0 m indigenous the bottom that the ramp.Use your dimensions of time and also the displacement to do a position vs. Time graph that the ball’s motion.Repeat procedures 4 v 6, through different world taking top top the duties of experimenter, timer, and recorder. Perform you gain the exact same measurement worths regardless of that releases the ball, procedures the time, or documents the result? Discuss feasible causes that discrepancies, if any.

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Grasp Check

True or False: The median speed the the ball will be less than the mean velocity of the ball.

True False

Solving difficulties Using position vs. Time Graphs

So just how do we usage graphs to fix for things we desire to recognize like velocity?