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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.

Graphs of motion where acceleration is not constant

Now consider the motion of the jet car as it goes from 165 m/s to its top velocity of 250 m/s, graphed in [link] . Time again starts at zero, and the initial displacement and velocity are 2900 m and 165 m/s, respectively. (These were the final displacement and velocity of the car in the motion graphed in [link] .) Acceleration gradually decreases from 5 . 0 m/s 2 to zero when the car hits 250 m/s. The slope of the x vs. t graph increases until t = 55 s size 12{t="55"`s} {} , after which time the slope is constant. Similarly, velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward.

Three line graphs of jet car displacement, velocity, and acceleration, respectively. First line graph is of position over time. Line is straight with a positive slope. Second line graph is of velocity over time. Line graph has a positive slope that decreases over time and flattens out at the end. Third line graph is of acceleration over time. Line has a negative slope that increases over time until it flattens out at the end. The line is not smooth, but has several kinks.
Graphs of motion of a jet-powered car as it reaches its top velocity. This motion begins where the motion in [link] ends. (a) The slope of this graph is velocity; it is plotted in the next graph. (b) The velocity gradually approaches its top value. The slope of this graph is acceleration; it is plotted in the final graph. (c) Acceleration gradually declines to zero when velocity becomes constant.

Calculating acceleration from a graph of velocity versus time

Calculate the acceleration of the jet car at a time of 25 s by finding the slope of the v size 12{v} {} vs. t size 12{t} {} graph in [link] (b).

Strategy

The slope of the curve at t = 25 s size 12{t="25"`s} {} is equal to the slope of the line tangent at that point, as illustrated in [link] (b).

Solution

Determine endpoints of the tangent line from the figure, and then plug them into the equation to solve for slope, a size 12{a} {} .

slope = Δ v Δ t = 260 m/s 210 m/s 51 s 1.0 s
a = 50 m/s 50 s = 1 . 0 m /s 2 .

Discussion

Note that this value for a is consistent with the value plotted in [link] (c) at t = 25 s size 12{t="25"`s} {} .

A graph of displacement versus time can be used to generate a graph of velocity versus time, and a graph of velocity versus time can be used to generate a graph of acceleration versus time. We do this by finding the slope of the graphs at every point. If the graph is linear (i.e., a line with a constant slope), it is easy to find the slope at any point and you have the slope for every point. Graphical analysis of motion can be used to describe both specific and general characteristics of kinematics. Graphs can also be used for other topics in physics. An important aspect of exploring physical relationships is to graph them and look for underlying relationships.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Sample chapters: openstax college physics for ap® courses. OpenStax CNX. Oct 23, 2015 Download for free at http://legacy.cnx.org/content/col11896/1.9
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