41)+Model+projectile+motion+with+parametric+equations.

**41) Model projectile motion with parametric equations.**
**This objective, just recently started in this period 1 class, has been increasingly fascinating to me. This topic includes modeling projectile motion, or motion paths created by objects that have been given horizontal momentum and verticle momentum that is acting agianst our earth's gravitational pull downwards. However, these models do not include air resistance in the calculations and equations used to model these paths. I find this topic incredibly interesting because we live around projectile motion, and interact with projectile motion throughout our lives. Even as children we have always been in awe at gravity, and its invisible effect to drag everything back down to the ground. This topic explores the math behind this seemingly bizarre phenomena, and we see this math everywhere; after all, Math is Life. **


 * What makes this topic particularly difficult to grasp for me, is the vertical constant of gravity. Gravity is unchanging, and will always pull downwards at a rate of 9.8 meters/second. But including this factor into our parametric equations that measure distance on the X-axis and the Y-axis, as functions of T, time; resulted in an equation that one must fully understand in order to use, and know what one is doing. **


 * The projectile motion model follows two specific parametric equations, each into their corresponding X1 and Y1 equation slot on the calculator, to create the projectile path. These equations are as follows: **
 * x = v 0 cost + x 0 **
 * y = v 0 sint - .5gt 2  + y 0 **


 * A breakdown of the equations: **


 * x = v 0 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">cost + x <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: sub;">0 **
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The first variable is v <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: sub;">0 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">, this is the velocity of the projectile at time, 0; or the initial velocity of the projectile.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The next variable is, Theda. This represents an unknown angle measure, the angle at which the projectile was launched at.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Lastly comes x <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: sub;">0 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">, this represents the horizontal starting point of the projectile at time 0.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">X is the resultant horizontal distance of these varibles.


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">y = v <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: sub;">0 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">sint - .5gt <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: super;">2 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> + y <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: sub;">0 **
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The first variable is v <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: baseline;">0 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">, this is the velocity of the projectile at time, 0; or the initial velocity of the projectile.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The next variable is, Theda. This represents an unknown angle measure, the angle at which the projectile was launched at.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Next is, - .5gt <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: super;">2 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">, this adds the downward force of g, gravity, over t, time; to the vertical motion of the projectile. (g = 9.8 m/s or 32 ft/s)
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"><span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Lastly is y <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 9px; text-decoration: none; vertical-align: sub;">0 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">, this is the vertical starting distance of the projectile from the ground.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Problems: **


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">You are in track, and you are throwing the javelin, you can throw it at 60 ft/s. The distance to beat is 110 ft, is it possible to beat the other team’s toss? **


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">A potato cannon that fires whole potatoes has a muzzle velocity of 1200 ft/s. What is the maximum range of the cannon if it is fired from 5 ft above the ground? How long with the potato be in the air before it splatters on the ground? **


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">A football is kicked into the air downfield. The velocity of the ball is 88 ft/s, the ball was kicked at an angle of 45 degrees. There is a constant gust of wind that is going against the ball in the opposite direction of the kick, the wind is moving at a speed of 18 ft/s. How long is the ball in the air, and how far will the ball go? **


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Portfolio Problem: **


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">A sniper is testing the effective range of a gun that is designed for an extreme distance of 8 km, the muzzle velocity is 853 m/s (the bullet fired from this gun remains perfectly staight for the duration of the bullet’s flight). He fires at a target that is 8 km away and that is 3 meters from the ground. At what angle must the shooter angle the gun, to hit the target while taking the effects of gravity into account? ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Video: **
 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|__http://www.youtube.com/watch?v=m7oLLbhzS2Q__] **
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The following is a video from Mythbusters, as they test the very concept of Projectile Motion: **


 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">__[]__ **