Discusses projectile motion as two dimensional motion. **More good stuff available at: www.wsautter.com and http://www.youtube.com/results?search_query=wnsautter&aq=f

1. PROJECTILES Motion in Two Dimensions Copyright Sautter 2011

2. Please let me promote some of my other work on the next slide and then the selected presentation will start. Thank you. Walt S.

3. I I have written six books: "Sticks - A Golfer's Tale" - A hacker's dream comes true. "Fish Farm" - Revenge of the old people. "Coach" - A mystery set a rural, bigoted, nineteen fifties football town. "The Three Dollar Phoenix" - A mystery set in Newark, New Jersey in the 1970s. "The Divine Comedy MMIX" - A humorous play about Jesus returning. "The Blood of Judas" - A horror story of revenge set in Nazi Germany. All are available at : smashwords .com or at : wsautter .com I have video trailers for "Coach", "Fish Farm" and "The Blood of Judas" at: youtube .com Please take a look. Thanks. Walt Sautter - wsautter @ optonline .net PS - Lots more stuff besides books at: wsautter .com

5. Projectile Motion JUST AFTER FIRING S y =0, S x = 0 V y = +max V x = constant A y = g, A x =0 AT HIGH POINT S y = max, S x = ½ max V y = 0, V x = constant A y = g, A x =0 JUST BEFORE LANDING S y = 0, S x = max V y = -max V x = constant A y = g, A x =0 CLICK HERE

6. A X COMPONENT Y COMPONENT X COMPONENT Y COMPONENT B Y COMPONENT X COMPONENT C Vector Components

7. S y V y inst = slope of tangents V y Time V y = 0 A y inst = slope of tangents A y Time A y inst = - 9.8 meters / sec 2 Projectile Motion (vertical component) Time

8. V o S x Path of object Without gravity S y = actual height of object Distance fallen due to gravity (1/2 g t 2 ) Vertical height if gravity did not act on the object (V o sin  t)  S y = height without effects of gravity – distance fallen due to gravity S y = V o sin  t + ½ g t 2 (the value of gravity is negative) Horizontal distance traveled by projectile S x = V o cos  t Vertical & Horizontal Displacements of a Projectile Actual projectile path Object Projected at Angle 

9. Water spraying from a hose is a common example of projectile motion The vertical motion of the water is accelerated by gravity. The horizontal motion is constant velocity and is unaffected by gravity! Velocity Components of Projectile Motion V y = V o sin + gt V x = V o cos  

10. Vertical displacements are equal for projectiles and dropped objects however horizontal displacements are greater for projectiles. Vertical acceleration for both dropped objects and projectiles is that of gravity (-32 ft/s 2 , -9.8 m/s 2 ). They both hit the ground at the exact same time but, of course, the projectile is further away ! Accelerated by gravity Projectile V x > 0 Constant velocity Vertical & Horizontal Motion of a Projectile vs. a Dropped Object Dropped Object V x = 0

11. Vertical & Horizontal Components of Motion during Projectile Flight V y = + max V y = - max V y = 0 Horizontal component of projectile motion (Constant velocity – no acceleration) Vertical component of projectile motion (Accelerated by gravity)

12. Projectile Motion Equations Vertical Displacement Horizontal Displacement Vertical Velocity Horizontal Velocity Vertical Acceleration Horizontal Acceleration a = g y a = o x y o S = V sin t + 1/2 g t 2  V = V cos x o  V = V sin + g t y o  S = V cos t x o 

13. Special Situation Projectile Equations Maximum Height Maximum Horizontal Distance (Range) h = - V sin max o 2 2 2 g -----------------  R = - V sin 2  2 ----------------- o g h max range

14. When an object is dropped at the exact same time a projectile is fired at the falling object, aiming directly at the object always insures a direct hit. Why ? Because the object and the projectile once fired, are in both in free fall ! Projectiles in Free Fall

21. Now it's time for you to try some problems on your own ! The problems are similar to the ones which have been solved so look back and review the appropriate problem if you get stuck !

24. Ideal vs. Real Projectile Paths Projectile Path without Air Resistance Projectile Path with Air Resistance

25. The End