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h = v0*t + 0.5*g*t^2

where

  • h is the distance of the projectile above the surface of the earth in units of distance
  • v0 is the initial velocity of the projectile in units of distance/time
  • t is time in seconds
  • g is the acceleration of gravity, approximately 9.8 meters per second squared, or approximately 32.2 feet per second squared.

Some physics textbooks also list the following equations as being important .

v = v0 + g*t

v^2 = v0^2 + 2*g*h

where v is the velocity of the object and the other terms are the same as described above .

Exercise to find the velocity

Let's do an exercise using the first of the two equations given above .

An individual on the surface of the earth shoots an arrow directly upward with a velocity of 100 feet per second. How many seconds elapse before the arrow turns and starts falling towards thesurface of the earth. Ignore the effects of air resistance.

Create a script

Please copy the code from Listing 6 into an html file and open the file in your browser.

Listing 6 . Exercise to find the velocity.
<!---------------- File JavaScript06.html ---------------------><html><body><script language="JavaScript1.3">document.write("Start Script<br/><br/>"); //Initialize the problem parameters.var g = -32.2;//gravity in ft/sec*sec on Earth var v0 = 100;//initial velocity in ft/sec//Given that v = v0 + g * t //At what time does the velocity go to zero?//Rearrange the terms in the equation. var t = -v0/g;//Display the results document.write("Arrow has zero velocity at " +t.toFixed(2) + " seconds " + "<br/>"); document.write("<br/>End Script");</script></body></html>

Screen output

The text shown in Figure 10 should appear in your browser window when you open the html file in your browser.

Figure 10 . Screen output for Listing #6.
Start Script Arrow has zero velocity at 3.11 secondsEnd Script

Analysis of the code

Compared to working through the solutions to the previous exercises, this one seems almost trivial.

After establishing values for the acceleration of gravity and the initial velocity of the arrow, the code in Listing 6 rearranges the first equation given above and solves for the value of time at which the velocity goes to zero. This is the point in time when the arrow turnsfrom moving up and begins falling back toward the earth.

The results are shown in Figure 10 . You should compare this result with Figure 1 , which shows that the arrow reaches its maximum height at approximately 3 seconds, which agrees very well with the result shown in Figure 10 .

Exercise to find the height

Let's do an exercise using the second of the two equations given above .

An individual that is six feet tall standing on the surface of the earth shoots an arrow directly upward with a velocity of 100 feet per second. What is the maximum height achieved by the arrow before it turns and falls backtowards the surface of the earth? Ignore the effects of air resistance.

Create a script

Please copy the code from Listing 7 into an html file and open the file in your browser.

Listing 7 . Exercise to find the height.
<!---------------- File JavaScript07.html ---------------------><html><body><script language="JavaScript1.3">document.write("Start Script<br/><br/>"); //Initialize the problem parameters.var g = -32.2;//gravity in ft/sec*sec on Earth var v0 = 100;//initial velocity in ft/secvar h0 = 6;//initial height //Given that v^2 = v0^2 +2*g*h //What is the maximum height reached by the arrow?//Note that the maximum height is six feet more than // the value given by the above equation because that// equation is based on the point of release. //The maximum height occurs when the velocity goes to zero.//Setting the velocity to zero and rearranging the terms // in the equation gives:var h = h0 + (-(v0 * v0))/(2*g); //Display the resultsdocument.write("Arrow reaches maximum height of " + h.toFixed(2) + " feet " + "<br/>"); document.write("<br/>End Script");</script></body></html>

Screen output

The text shown in Figure 11 should appear in your browser window when the html file is opened in your browser.

Figure 11 . Screen output for Listing #7.
Start Script Arrow reaches maximum height of 161.28 feetEnd Script

Analysis of the code

Once again, compared to working through the previous exercises, this one also seems almost trivial.

After establishing values for the acceleration of gravity, the initial velocity of the arrow, and the height at which the arrow was released, the code in Listing 7 rearranges the second equation given above and solves for the value of the height (relative to the release point) at which the velocity goes to zero. This is the pointin the trajectory where the arrow turns from moving up and begins falling back toward the earth.

Note that in order to get the actual height, it was necessary to add the initial height of 6 feet to the computed height.

Compare the results

The results are shown in Figure 11 . You should compare this result with Figure 2 , which shows that the arrow reaches its maximum height at approximately 161.1 feet, which agrees very well with the result shown in Figure 11 .

Run the scripts

I encourage you to run the scripts that I have presented in this lesson to confirm that you get the same results. Copy the code for each script into atext file with an extension of html. Then open that file in your browser. Experiment with the code, making changes, and observing the results of your changes. Makecertain that you can explain why your changes behave as they do.

Miscellaneous

This section contains a variety of miscellaneous information.

Housekeeping material
  • Module name: GAME 2302 Motion -- Variable Velocity and Acceleration
  • File: Game0380.htm
  • Published: 10/13/12
  • Revised: 02/01/16
Disclaimers:

Financial : Although the Connexions site makes it possible for you to download a PDFfile for this module at no charge, and also makes it possible for you to purchase a pre-printed version of the PDF file, you should be aware thatsome of the HTML elements in this module may not translate well into PDF.

I also want you to know that, I receive no financial compensation from the Connexions website even if you purchase the PDF version of the module.

In the past, unknown individuals have copied my modules from cnx.org, converted them to Kindle books, and placed them for sale on Amazon.comshowing me as the author. I neither receive compensation for those sales nor do I know who does receive compensation. If you purchase such a book, pleasebe aware that it is a copy of a module that is freely available on cnx.org and that it was made and published without my prior knowledge.

Affiliation : I am a professor of Computer Information Technology at Austin Community College in Austin, TX.

-end-

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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what's the easiest and fastest way to the synthesize AgNP?
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China
Cied
types of nano material
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Source:  OpenStax, Game 2302 - mathematical applications for game development. OpenStax CNX. Jan 09, 2016 Download for free at https://legacy.cnx.org/content/col11450/1.33
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