Position vs. Time Graphs, Motion in One Dimension | Physics

Learning objectives

Section Key Terms

dependent variable

independent variable

tangent

Graphing Position as a Function of Time

A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information, they also 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 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, you would have referred to the horizontal axis as the x-axis and the vertical axis as the y-axis. As in Figure_02_03_graph, a straight-line graph has the general form $y = m x + b$ .

Here m is the slope, defined as the rise 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 real world, these quantities will take on a specific significance, as we will see below. (Figure_02_03_graph.)

View original figure (external; image not reproduced in this text edition)

Original figure description: 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.

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

In physics, time is usually the independent variable. Other quantities, such as displacement, are said to depend upon it. A graph of position versus time, therefore, would have position on the vertical axis (dependent variable) and time on the horizontal axis (independent variable). In this case, to what would the slope and y-intercept refer? Let’s look back at our original example when studying distance and displacement.

The drive to school was 5 km from home. Let’s assume it 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 like that shown in Figure_02_03_position.

View original figure (external; image not reproduced in this text edition)

Original figure description: 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.

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?

As we said before, d₀ = 0 because we call home our O and start calculating from there. In Figure_02_03_position, the line starts at d = 0, as well. This is the b in our equation for a straight line. Our initial position in a position versus time graph is always the place where the graph crosses the x-axis at t = 0. What is the slope? The rise is the change in position, (i.e., displacement) and the run is the change in time. This relationship can also be written

\frac{Δ d}{Δ t} .

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

Let’s look at another example. Figure_02_03_time shows a graph of position versus time for a jet-powered car on a very flat dry lake bed in Nevada.

View original figure (external; image not reproduced in this text edition)

Original figure description: 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.

The diagram shows a graph of position versus time for a jet-powered car on the Bonneville Salt Flats.

Using the relationship between dependent and independent variables, we see that the slope in the graph in Figure_02_03_time is average velocity, v_avg and the intercept is displacement at time zero—that is, d₀. Substituting these symbols into y = mx + b gives

d = v t + d_{0}

d = d_{0} + v t .

Thus a graph of position versus time gives a general relationship among displacement, velocity, and time, as well as 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 m at t = 1.0 s, and so on. And we can learn about the object’s velocity, as well.

Teacher Support

Help students learn what different graphs of displacement vs. time look like.

[Visual] Set up a meter stick.

If you can find a remote control car, have one student record times as you send the car forward along the stick, then backwards, then forward again with a constant velocity.
Take the recorded times and the change in position and put them together.
Get the students to coach 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 as zero affects the graph.

[AL] Is it realistic to draw any position graph that starts at rest without some curve in it? Why might we 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 displacement, but they can also use the graph to determine the total 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 direction 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 learn about in the next chapter.

Graphing Motion

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

Choose an open location with lots of space to spread out so there is less chance for tripping or falling due to rolling balls.

1 ball
1 board
2 or 3 books
1 stopwatch
1 tape measure
6 pieces of masking tape
1 piece of graph paper
1 pencil

Procedure

Build a ramp by placing one end of the board on top 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 3 m.
Mark distances 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. Write the distances on the tape.
Have one person take the role of the experimenter. This person will release the ball from the top of the ramp. If the ball does not 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 release the ball. Have a second person, the timer, begin timing the trial once the ball reaches the bottom of the ramp and stop the timing once the ball reaches 0.5 m. Have a third person, the recorder, record the time in a data table.
Repeat Step 4, stopping the times at the distances of 1.0 m, 1.5 m, 2.0 m, 2.5 m, and 3.0 m from the bottom of the ramp.
Use your measurements of time and the displacement to make a position vs. time graph of the ball’s motion.
Repeat Steps 4 through 6, with different people taking on the roles of experimenter, timer, and recorder. Do you get the same measurement values regardless of who releases the ball, measures the time, or records the result? Discuss possible causes of discrepancies, if any.

Source link

Teacher Support

[BL][OL] Emphasize that the motion in this lab is the motion of the ball as it rolls along the floor. Ask students where there zero should be.

[AL] Ask students what the graph would look like if they began timing at the top versus the bottom of the ramp. Why would the graph look different? What might account for the difference?

[BL][OL] Have the students compare the graphs made 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? Do the differences appear to be random, or are there systematic differences? Why might there be systematic differences between the two sets of measurements with different individuals in each role?

Solving Problems Using Position vs. Time Graphs

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

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

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

Solution

Choose two points on the line. In this case, we choose the points labeled on the graph: (6.4 s, 2000 m) and (0.50 s, 525 m). (Note, however, that you could choose any two points.)
Substitute the d and t values of the chosen points into the equation. Remember in calculating change (Δ) we always use final value minus initial value. $\begin{matrix} v & = & \frac{Δ d}{Δ t} \\ = & \frac{2000 m - 525 m}{6.4 s - 0.50 s} \\ = & 250 m/s \end{matrix},$

Discussion

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

But what if the graph of the position is more complicated than a straight line? What if the object speeds up or turns around and goes backward? Can we figure out anything about its velocity from a graph of that kind of motion? Let’s take another look at the jet-powered car. The graph in Figure_02_03_jet shows its motion as it is getting up to speed after starting at rest. Time starts at zero for this motion (as if measured with a stopwatch), and the displacement and velocity are initially 200 m and 15 m/s, respectively.

View original figure (external; image not reproduced in this text edition)

Original figure description: 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.

The diagram shows a graph of the position of a jet-powered car during the time span when it is speeding up. The slope of a distance versus time graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that point.

View original figure (external; image not reproduced in this text edition)

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

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

The graph of position versus time in Figure_02_03_jet is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a position-versus-time graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in Figure_02_03_jet. The average velocity is the net displacement divided by the time traveled.

Using Position–Time Graph to Calculate Average Velocity: Jet Car, Take Two

Calculate the instantaneous velocity of the jet car at a time of 25 s by finding the slope of the tangent line at point Q in Figure_02_03_jet.

Solution

Find the tangent line to the curve at $t = 25 s$ .
Determine the endpoints of the tangent. These correspond to a position of 1,300 m at time 19 s and a position of 3120 m at time 32 s.
Plug these endpoints into the equation to solve for the slope, v. $\begin{matrix} slope & = & v_{Q} = \frac{Δ d_{Q}}{Δ t_{Q}} \\ = & \frac{(3120 - 1300) m}{(32 - 19) s} \\ = & \frac{1820 m}{13 s} \\ = & 140 m/s \end{matrix}$

Discussion

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

Practice Problems

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

View original figure (external; image not reproduced in this text edition)

Original figure description: The image depicts a graph titled 'position vs time' with position (m) on the y-axis and time (s) on the x-axis. The data points graphed trend upward in an evenly distributed straight line from left to right.

0.25 m/s
0.31 m/s
3.2 m/s
4.00 m/s

Source link

Check Your Understanding

Which of the following information about motion can be determined by looking at a position vs. time graph that is a straight line?

frame of reference
average acceleration
velocity
direction of force applied

Source link

Section Summary

Graphs can be used to analyze motion.
The slope of a position vs. time graph is the velocity.
For a straight line graph of position, the slope is the average velocity.
To obtain the instantaneous velocity at a given moment for a curved graph, find the tangent line at that point and take its slope.

Key Equations

Displacement

d = d_{0} + v t

Concept Items

Use the graph to describe what the runner’s motion looks like.

View original figure (external; image not reproduced in this text edition)

Original figure description: A graph titled Runner Position plots time (s) on the x axis, and position (m) on the y axis. A line extends from the origin with a slope of 3. At point (12, 4), it extends without a slope and points (12, 6), (12, 8), and (12, 10) are marked.

How are average velocity for only the first four seconds and instantaneous velocity related? What is the runner's net displacement over the time shown?

The net displacement is 12 m and the average velocity is equal to the instantaneous velocity.
The net displacement is 12 m and the average velocity is two times the instantaneous velocity.
The net displacement is 10 + 12 = 22 m and the average velocity is equal to the instantaneous velocity.
The net displacement is 10 + 12 = 22 m and the average velocity is two times the instantaneous velocity.

Source link

Critical Thinking

Explain how you can use the graph of position vs. time to describe the change in velocity over time.

View original figure (external; image not reproduced in this text edition)

Original figure description: A position-time graph (x vs. t) showing an object's motion with decreasing positive velocity from points a to c, reaching zero velocity at d, and remaining stationary at e.

Identify the time (t_a, t_b, t_c, t_d, or t_e) at which at which the instantaneous velocity is greatest, the time at which it is zero, and the time at which it is negative.

Problems

Using the graph, what is the average velocity for the whole 10 seconds?

View original figure (external; image not reproduced in this text edition)

The total average velocity is 0 m/s.
The total average velocity is 1.2 m/s.
The total average velocity is 1.5 m/s.
The total average velocity is 3.0 m/s.

Source link

Test Prep Multiple Choice

What is the slope of a straight line graph of position vs. time?

Velocity
Displacement
Distance
Acceleration

Using the graph, what is the runner’s velocity from 4 to 10 s?

View original figure (external; image not reproduced in this text edition)

Original figure description: A line graph titled 'Runner Position' shows a runner's position (m) over time (s). The runner moves from 0m to 12m in the first 4 seconds, then remains at 12m from 4 to 10 seconds.

–3 m/s
0 m/s
1.2 m/s
3 m/s

Test Prep Short Answer

A hockey puck is shot down the arena in a straight line. Assume it does not slow until it is stopped by an opposing player who sends it back in the direction it came. The players are 20 m apart and it takes 1 s for the puck to go there and back. Which of the following describes the graph of the displacement over time? Consider the initial direction of the puck to be positive.

The graph is an upward opening V.
The graph is a downward opening V.
The graph is an upward opening U.
The graph is downward opening U.

A defensive player kicks a soccer ball 20 m back to her own goalie. It stops just as it reaches her. She sends it back to the player. Without knowing the time it takes, draw a rough sketch of the displacement over time. Does this graph look similar to the graph of the hockey puck from the previous question?

Yes, the graph is similar to the graph of the hockey puck.
No, the graph is not similar to the graph of the hockey puck.
The graphs cannot be compared without knowing the time the soccer ball was rolling.

What are the net displacement, total distance traveled, and total average velocity in the previous two problems?

net displacement = 0 m, total distance = 20 m, total average velocity = 20 m/s
net displacement = 0 m, total distance = 40 m, total average velocity = 20 m/s
net displacement = 0 m, total distance = 20 m, total average velocity = 0 m/s
net displacement = 0 m, total distance = 40 m, total average velocity = 0 m/s

Source link

Test Prep Extended Response

What kind of motion would create a position graph like the one shown?

View original figure (external; image not reproduced in this text edition)

Original figure description: A position versus time graph illustrates an object's increasing position over time, with the red line indicating an accelerating motion as the slope becomes steeper.

uniform motion
any motion that accelerates
motion that stops and then starts
motion that has constant velocity

What is the average velocity for the whole time period shown in the graph?

View original figure (external; image not reproduced in this text edition)

Original figure description: A position-time graph showing an object's position in meters (y-axis) versus time in seconds (x-axis). The object moves from 0m to 2m, then to -3m, stays at -3m, and finally moves to -2m.

$- \frac{1}{3} m/s$
$- \frac{3}{4} m/s$
$\frac{1}{3} m/s$
$\frac{3}{4} m/s$

Glossary

dependent variable: the variable that changes as the independent variable changes
independent variable: the variable, usually along the horizontal axis of a graph, that does not change based on human or experimental action; in physics this is usually time
tangent: a line that touches another at exactly one point

Attribution. Physics, OpenStax, Rice University, and the contributing authors credited in the book. Licensed under Creative Commons Attribution 4.0 International. Source revision 56e71824e7 of openstax/osbooks-physics (2026-10-13). CC BY 4.0. Original source ↗. Text edition of a pinned CC BY 4.0 source revision of the OpenStax book, converted from CNXML by the EvidenceHelix importer. Original prose, learning objectives, tables, exercises, glossary and captions are preserved; images are not copied (figures link to the original source and keep their credits). No new content was added. Later revisions of this book can carry different licence terms and are not used.