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Additional general information can be obtained from [link] and the expression for a straight line, y = mx + b size 12{y= ital "mx"+b} {} .

In this case, the vertical axis y size 12{y} {} is V size 12{V} {} , the intercept b size 12{b} {} is v 0 size 12{v rSub { size 8{0} } } {} , the slope m size 12{m} {} is a size 12{a} {} , and the horizontal axis x size 12{x} {} is t size 12{t} {} . Substituting these symbols yields

v = v 0 + at . size 12{v=v rSub { size 8{0} } + ital "at"} {}

A general relationship for velocity, acceleration, and time has again been obtained from a graph. Notice that this equation was also derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension .

It is not accidental that the same equations are obtained by graphical analysis as by algebraic techniques. In fact, an important way to discover physical relationships is to measure various physical quantities and then make graphs of one quantity against another to see if they are correlated in any way. Correlations imply physical relationships and might be shown by smooth graphs such as those above. From such graphs, mathematical relationships can sometimes be postulated. Further experiments are then performed to determine the validity of the hypothesized relationships.

Section summary

  • Graphs of motion can be used to analyze motion.
  • Graphical solutions yield identical solutions to mathematical methods for deriving motion equations.
  • The slope of a graph of displacement x size 12{x} {} vs. time t size 12{t} {} is velocity v size 12{v} {} .
  • The slope of a graph of velocity v size 12{v} {} vs. time t size 12{t} {} graph is acceleration a size 12{a} {} .
  • Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.

Conceptual questions

(a) Explain how you can use the graph of position versus time in [link] to describe the change in velocity over time. Identify (b) the time ( t a , t b , t c , t d , or t e ) at which the instantaneous velocity is greatest, (c) the time at which it is zero, and (d) the time at which it is negative.

Line graph of position versus time with 5 points labeled: a, b, c, d, and e. The slope of the line changes. It begins with a positive slope that decreases over time until around point d, where it is flat. It then has a slightly negative slope.

(a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in [link] . (b) Identify the time or times ( t a , t b , t c , etc.) at which the instantaneous velocity is greatest. (c) At which times is it zero? (d) At which times is it negative?

Line graph of position over time with 12 points labeled a through l. Line has a negative slope from a to c, where it turns and has a positive slope till point e. It turns again and has a negative slope till point g. The slope then increases again till l, where it flattens out.

(a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in [link] . (b) Based on the graph, how does acceleration change over time?

Line graph of velocity over time with two points labeled. Point P is at v 1 t 1. Point Q is at v 2 t 2. The line has a positive slope that increases over time.

(a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in [link] . (b) Identify the time or times ( t a , t b , t c , etc.) at which the acceleration is greatest. (c) At which times is it zero? (d) At which times is it negative?

Line graph of velocity over time with 12 points labeled a through l. The line has a positive slope from a at the origin to d where it slopes downward to e, and then back upward to h. It then slopes back down to point l at v equals 0.

Consider the velocity vs. time graph of a person in an elevator shown in [link] . Suppose the elevator is initially at rest. It then accelerates for 3 seconds, maintains that velocity for 15 seconds, then decelerates for 5 seconds until it stops. The acceleration for the entire trip is not constant so we cannot use the equations of motion from Motion Equations for Constant Acceleration in One Dimension for the complete trip. (We could, however, use them in the three individual sections where acceleration is a constant.) Sketch graphs of (a) position vs. time and (b) acceleration vs. time for this trip.

Line graph of velocity versus time. Line begins at the origin and has a positive slope until it reaches 3 meters per second at 3 seconds. The slope is then zero until 18 seconds, where it becomes negative until the line reaches a velocity of 0 at 23 seconds.

A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the cylinder vs. time as it goes up and then down the plane.

Problems&Exercises

Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.

(a) By taking the slope of the curve in [link] , verify that the velocity of the jet car is 115 m/s at t = 20 s size 12{t="20"`s} {} . (b) By taking the slope of the curve at any point in [link] , verify that the jet car’s acceleration is 5 . 0 m/s 2 size 12{5 "." "0 m/s" rSup { size 8{2} } } {} .

Line graph of position over time. Line has positive slope that increases over time.
Line graph of velocity versus time. Line is straight with a positive slope.

(a) 115 m/s size 12{"115 m/s"} {}

(b) 5 . 0 m/s 2 size 12{5 "." "0 m/s" rSup { size 8{2} } } {}

Construct the displacement graph for the subway shuttle train as shown in [link] (a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the information on acceleration and velocity given in the examples for this figure.

Line graph of position versus time. Line begins with a slight positive slope. It then kinks to a much greater positive slope.
Practice Key Terms 4

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Source:  OpenStax, Kinematics. OpenStax CNX. Sep 11, 2015 Download for free at https://legacy.cnx.org/content/col11878/1.5
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