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The magnitude of the average velocity vector, identified by the variable named vecVelMag , is compute by dividing the magnitude of the displacement vector by the total time. This results in a value havingunits of feet/second as shown by the comments.

The direction of the average velocity vector

The direction of the average velocity vector, identified by the variable named vecVelAng , is recognized as being the same as the direction of the overall displacement vector with units of degrees.

Display the results

Finally, the document.write method is called several times in succession to display theoutput text shown in Figure 3 .

Multiple concurrent velocities

When a body has multiple concurrent velocities, the overall velocity of the body is equal to the vector sum of the individual velocities. You can computethat vector sum in any way that works for you, including the parallelogram rule, a tail-to-head vector diagram, or the mathematical techniques that we will usehere.

Relative velocity

In this section, we will also be dealing with a topic called relative velocity . For example, suppose we see a man walking down the aisle in a passenger car of a train that is moving slowly at a uniform velocity along astraight track. How fast is the man moving?

The frame of reference

The answer to that question depends on the frame of reference of the observer. For example, to another passenger in the same rail car, it may appear that the man is moving at about3 feet per second, which is a reasonably comfortable walking speed.

However, to someone standing on the ground outside of the passenger car (pretend that the side of the car is transparent), it may appearthat the man is moving much faster or much slower than 3 feet per second, depending on which direction the man is moving relative to the motion of thetrain.

It could even turn out that insofar as the outside observer is concerned, the man isn't moving at all, or is moving backwards.

Exercise #1 for a man on a train

A man is walking along the aisle in a passenger car of a train that is moving at 1 mile per hour toward the east. The man is walking in the same direction thatthe train is moving. The man is walking at a uniform velocity of 2.933 feet per second. (You will see shortly why I chose such a strange walking speed for the man.) What is the man's overall velocitywith reference to the ground?

Figure 4 shows a vector diagram for the man onthe train. The units for both velocity vectors must be the same. Therefore, the length ofthe velocity vector for the man is based on a conversion from 2.933 feet per second to 2 miles per hour.

Figure 4 - Vector diagram for a man on a train.

Missing image

JavaScript code

Please copy the code shown in Listing 3 into an html file and open the file in your browser.

Listing 3 . Exercise #1 for a man on a train.

<!---------------- File JavaScript03.html ---------------------><html><body><script language="JavaScript1.3">document.write("Start Script<br/>"); //The purpose of this function is to receive the adjacent// and opposite side values for a right triangle and to // return the angle in degrees in the correct quadrant.function getAngle(x,y){ if((x == 0)&&(y == 0)){ //Angle is indeterminate. Just return zero.return 0; }else if((x == 0)&&(y>0)){ //Avoid divide by zero denominator.return 90; }else if((x == 0)&&(y<0)){ //Avoid divide by zero denominator.return -90; }else if((x<0)&&(y>= 0)){ //Correct to second quadrantreturn Math.atan(y/x)*180/Math.PI + 180; }else if((x<0)&&(y<= 0)){ //Correct to third quadrantreturn Math.atan(y/x)*180/Math.PI + 180; }else{//First and fourth quadrants. No correction required. return Math.atan(y/x)*180/Math.PI;}//end else }//end function getAngle//------------------------------------------------------------// //The purpose of this function is to add two vectors, given// the magnitude and angle (in degrees) for each vector. // The magnitude and angle (in degrees) of the resultant// vector is returned in a two-element array, with the // magnitude in the element at index 0 and the angle in the// element at index 1. Note that this function calls the // getAngle function, which must also be provided in the// script. To use this function to subtract one vector from // another, add 180 degrees to the angle for the subtrahend// vector before passing the angle to the function. To use // this function to add more than two vectors, add the first// two vectors as normal, then add the output from this // function to the third vector, etc., until all of the// vectors have been included in the sum. function vectorSum(vecAmag,vecAang,vecBmag,vecBang){var vecResult = new Array(2); //Compute the horizontal and vertical components// of each vector. var vecAh = vecAmag*Math.cos(vecAang*Math.PI/180);var vecAv = vecAmag*Math.sin(vecAang*Math.PI/180); var vecBh = vecBmag*Math.cos(vecBang*Math.PI/180);var vecBv = vecBmag*Math.sin(vecBang*Math.PI/180); //Compute the sums of the horizontal and vertical// components from the two vectors to get the // horizontal and vertical component of the// resultant vector. var vecResultH = vecAh + vecBh;var vecResultV = vecAv + vecBv; //Use the Pythagorean theorem to compute the magnitude of// the resultant vector. vecResult[0]= Math.sqrt(Math.pow(vecResultH,2) + Math.pow(vecResultV,2));//Compute the angle of the resultant vector in degrees. vecResult[1]= getAngle(vecResultH,vecResultV); return vecResult;}//end vectorSum function //------------------------------------------------------------////The main script body begins here. //Establish the magnitude and angle for each vector.var vecTrainMag = 1;//magnitude of velocity of train in miles/hr var vecTrainAng = 0;//angle of velocity of train in degreesvar vecManMag = 2.933*3600*(1/5280);//Magnitude of velocity of // man (ft/sec)*(sec/hr)*(mile/ft)// = miles/hour var vecManAng = 0;//angle of velocity of man in degrees//Add the two vectors var resultant = vectorSum(vecTrainMag,vecTrainAng,vecManMag,vecManAng); //Display the magnitude and direction of the resultant vector.document.write("Velocity magnitude = " + resultant[0].toFixed(2) + " miles/hour<br/>"); document.write("Velocity angle = " +resultant[1].toFixed(2) + " degrees<br/>"); document.write("End Script");</script></body></html>

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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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