# 2.8 Game 2302-0380 motion -- variable velocity and acceleration  (Page 17/17)

 Page 17 / 17

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-

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