1. Brain Teaser During the Little League baseball season, every team plays every other team in the league ten times. If there are ten teams in the league, how many games are played in the league in one season?

5. Projectiles Launched at an Angle Ms. Bartlett November 12-13, 2015

6. Projectiles Launched at an Angle Demonstration: Tossing a Ball If the object is launched upward, like a ball tossed straight up in the air, it rises with slowing speed, reaches the top of its path, and descends with increasing speed. A projectile launched at an angle would continue in a straight line at a constant velocity if gravity is ignored. However, gravity makes the projectile accelerate to Earth. Notice the projectile follows a parabolic trajectory.

7. Projectiles Launched at an Angle Since the projectile is launched at an angle, it now has both horizontal and vertical velocities. The horizontal component of the velocity remains constant. The vertical component of the velocity changes as the projectile moves up or down.

8. Projectiles Launched at an Angle The up and right vectors represent the velocity given to the projectile when launched. The vertical vectors decrease in magnitude due to gravity. Eventually, the effects of gravity will reduce the upward velocity to zero. This occurs at the top of the parabolic trajectory where there is only horizontal motion.

9. Projectiles Launched at an Angle After gravity reduces the upward (vertical) speed to zero it begins to add a downward velocity. This velocity increases until the projectile return to the ground.

10. Projectiles Launched at an Angle When looking at each half of the trajectory (up and down) you can determine that the speed of the projectile going up is equal to the speed of the projectile coming down (provided air resistance is ignored). The only difference is the direction of the motion. The other quantity depicted is the range, which is the horizontal distance that the projectile travels. Not shown is the flight time, which is how much time the projectile is in the air. For football punts, flight time often is called hang time.

11. Projectiles Launched at an Angle Notice the x and y components of the velocity vector as the golf ball travels along its parabolic path.

12. Projectiles Launched at an Angle Maximum range is achieved with a projection angle of 45°. For projection angles above and below 45°, the range is shorter, and it is equal for angles equally different from 45° (for example, 30° and 60°).

13. Projectile Motion So far, air resistance has been ignored in the analysis of projectile motion. While the effects of air resistance are very small for some projectiles, for others, the effects are large and complex. For example, dimples on a golf ball reduce air resistance and maximize its range. The force due to air resistance does exist and it can be important.

14. Problem Solving with Projectile Motion Sketch the problem. List givens and unknowns. Divide the projectile motion into a vertical motion problem and a horizontal motion problem. The vertical motion of a projectile is exactly that of an object dropped or thrown straight up or down with constant acceleration g. Use your constant acceleration (kinematics) equations. The horizontal motion of a projectile is the same as solving a constant velocity problem. Use d x = v x t and v x i = v x f. Vertical and horizontal motion are connected through the variable time.

15. Projectile Motion Review Formulas: d y = v i t + ½ a t 2 constant acceleration in the y- direction d x = v x tconstant velocity in the x- direction t = √(2d y )for a projectile that is launched horizontally, the g time only depends on the height Key Point: In projectile motion, the vertical and horizontal components of motion are independent.

16. Projectile Motion Project Driving Question: To what extent can you calculate with precision the time and average horizontal and vertical velocities of a projectile launched at a waste basket 2 meters away? Note: Must do this at least 20 times for accuracy and precision. Start planning with your #5 Boron Partner! Write the beginning of your Lab Report!

17. Problems #6-8 Homework! Due next class!