# 2.8 Graphical analysis of one-dimensional motion  (Page 9/8)

 Page 1 / 8
• Describe a straight-line graph in terms of its slope and y -intercept.
• Determine average velocity or instantaneous velocity from a graph of position vs. time.
• Determine average or instantaneous acceleration from a graph of velocity vs. time.
• Derive a graph of velocity vs. time from a graph of position vs. time.
• Derive a graph of acceleration vs. time from a graph of velocity vs. time.

A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information; they also reveal relationships between physical quantities. This section uses graphs of displacement, velocity, and acceleration versus time to illustrate one-dimensional kinematics.

## Slopes and general relationships

First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an independent variable    and the vertical axis a dependent variable    . If we call the horizontal axis the $x$ -axis and the vertical axis the $y$ -axis, as in [link] , a straight-line graph has the general form

$y=\text{mx}+b.$

Here $m$ is the slope    , defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter $b$ is used for the y -intercept , which is the point at which the line crosses the vertical axis.

## Graph of displacement vs. time ( a = 0, so v Is constant)

Time is usually an independent variable that other quantities, such as displacement, depend upon. A graph of displacement versus time would, thus, have $x$ on the vertical axis and $t$ on the horizontal axis. [link] is just such a straight-line graph. It shows a graph of displacement versus time for a jet-powered car on a very flat dry lake bed in Nevada.

Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity $\stackrel{-}{v}$ and the intercept is displacement at time zero—that is, ${x}_{0}$ . Substituting these symbols into $y=\text{mx}+b$ gives

$x=\stackrel{-}{v}t+{x}_{0}$

or

$x={x}_{0}+\stackrel{-}{v}t.$

Thus a graph of displacement versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information about a specific situation.

## The slope of x Vs. t

The slope of the graph of displacement $x$ vs. time $t$ is velocity $v$ .

$\text{slope}=\frac{\Delta x}{\Delta t}=v$

Notice that this equation is the same as that derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension .

From the figure we can see that the car has a displacement of 25 m at 0.50 s and 2000 m at 6.40 s. Its displacement at other times can be read from the graph; furthermore, information about its velocity and acceleration can also be obtained from the graph.

## Determining average velocity from a graph of displacement versus time: jet car

Find the average velocity of the car whose position is graphed in [link] .

Strategy

The slope of a graph of $x$ vs. $t$ is average velocity, since slope equals rise over run. In this case, rise = change in displacement and run = change in time, so that

$\text{slope}=\frac{\Delta x}{\Delta t}=\stackrel{-}{v}.$

Since the slope is constant here, any two points on the graph can be used to find the slope. (Generally speaking, it is most accurate to use two widely separated points on the straight line. This is because any error in reading data from the graph is proportionally smaller if the interval is larger.)

Solution

1. 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.)

2. Substitute the $x$ and $t$ values of the chosen points into the equation. Remember in calculating change $\left(\Delta \right)$ we always use final value minus initial value.

$\stackrel{-}{v}=\frac{\Delta x}{\Delta t}=\frac{\text{2000 m}-\text{525 m}}{6\text{.}\text{4 s}-0\text{.}\text{50 s}},$

yielding

$\stackrel{-}{v}=\text{250 m/s}.$

Discussion

This is an impressively large land speed (900 km/h, or about 560 mi/h): much greater than the typical highway speed limit of 60 mi/h (27 m/s or 96 km/h), but considerably shy of the record of 343 m/s (1234 km/h or 766 mi/h) set in 1997.

#### Questions & Answers

what is mass
is the amount of an object
Sendawula
mass is the measure of the inertia of a body
Ishmeal
advantages of CRO over ordinary voltmeter
what is the difference between displacement and distance?!
what is equilibrium
If a system is said to be under equilibrium whenever there is no force act upon it... And it remain in its initial stage..
soniya
what is velocity
Ahmed
What is conductivity
It is the ease with which electrical charges or heat can be transmitted through a material or a solution.
Cffrrcvccgg
how to find magnitude and direction
how to caclculate for speed
Arjune
derivation of ohms law
derivation of resistance
Kazeem
R=v/I where R=resistor, v=voltage, I=current
Kazeem
magnitude
Arjune
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
Electric current is the flow of electrons
is there really flow of electrons exist?
babar
Yes It exists
Cffrrcvccgg
explain plz how electrons flow
babar
if electron flows from where first come and end the first one
babar
an electron will flow accross a conductor because or when it posseses kinectic energy
Cffrrcvccgg
electron can not flow jist trasmit electrical energy
ghulam
free electrons of conductor
ankita
electric means the flow heat current.
electric means the flow of heat current in a circuit.
Serah
What is electric
electric means?
ghulam
electric means the flow of heat current in a circuit.
Serah
electric means the flow of electric current through conductor
the continuos flow of electrons in a circuit is called electric
ANUBHA
electric means charge
ghulam
electric means current
flow of current.
Sendawula
a boy cycles continuously through a distance of 1.0km in 5minutes. calculate his average speed in ms-1(meter per second). how do I solve this
speed = distance/time be sure to convert the km to m and minutes to seconds check my utube video "mathwithmrv speed"
PhysicswithMrV
d=1.0km÷1000=0.001 t=5×60=300s s=d\t s=0.001/300=0.0000033m\s
Serah
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
D=1km=1000m t=5mins×60secs=300sec s=d/t=3.333m/s
Daniel
I think Daniel Glorious is ryt
Amalia
why we cannot use DC instead of AC in a transformer
becuse the d .c cannot travel for long distance trnsmission
ghulam
what is physics
branch of science which deals with matter energy and their relationship between them
ghulam
Life science
the
what is heat and temperature
how does sound affect temperature
sound is directly proportional to the temperature.
juny