Projectile Motion: 2D Kinematics

1. Unit 4: Two-Dimensional Kinematics

2. Section A: Projectile Motion • Corresponding Book Sections: • 4.1, 4.2 • PA Assessment Anchors • S11.C.3.1

3. Difference between 1-D and 2-D • One Dimension • Up / Down • Back / Forth • Left / Right • Example: • Driving a car down a straight street • Two dimension • Projectiles • Vertical & Horizontal motion • Example: • Throwing something up in the air to someone else

4. Projectile Motion • Motion of objects that are launched • Objects continue moving under only the influence of gravity.

5. Basic assumptions of this unit… 1. Horizontal and Vertical motions are independent • In other words…treat the horizontal motion as if the vertical motion weren’t there, and vice-versa • You may need to use quantities in both directions, but you treat them separately (i.e.: Separate equations)

6. Basic assumptions of this unit… 2. Ignore air resistance • We all know that air resistance exists, but to make our lives easier, we’re going to ignore it • Otherwise, the problems get too hard!!

7. Basic assumptions of this unit… 3. We also ignore the rotation of the Earth • If we were to include the rotation of the Earth, we’d need to include that force in all of the problems…and why would we want to do that? 

8. Basic assumptions of this unit… 4. The acceleration of gravity is always 9.8 m/s2 and pulls in the downward direction • This is the same from the last unit. Just remember, if: • You say ↑ is positive, g is negative • You say ↑ is negative, g is positive

9. Basic assumptions of this unit… 5. Gravity only affects the motion in the y-direction and has no effect on the x-direction. • Think about it…if we’re analyzing the motion separately (vertical and horizontal), when we look at the horizontal motion, gravity doesn’t affect that motion.

10. The basic kinematics equations… 2-D

11. Getting Components for the Equations • The equations are the same, they just analyze the x and y directions separately • Remember from vectors: Ax = A cos θ Ay = A sin θ vox = vocosθ voy = vosinθ so......

12. Two ways to solve the turtle problem... Method #1 Using vector principles 1 m Problem: How far has the turtle traveled in 5 s (both x and y dir)?

13. Two ways to solve the turtle problem... Method #2 Using kinematics equations = .2 m/s Problem: How far has the turtle traveled in 5 s (both x and y dir)?

14. Practice Problem #1 • Refer to Example 4-1 on page 79 Skip 2009-2010

15. Practice Problem #2 • Refer to Example 4-2 on Page 80 Skip 2009-2010

16. Section B: Zero Launch Angle • Corresponding Book Sections: • 4.3 • PA Assessment Anchors • S11.C.3.1

17. Zero Launch Angle • Projectile launched horizontally • In other words, the angle between initial velocity and horizontal is 0° • Projectile has no acceleration in the x-direction unless specified • Initial velocity is only in x-direction.

18. Practice Problem #1 • A person is walking with a speed of 1.3 m/s and drops a ball he is holding. The ball falls from a height of 1.25 m. Find the horizontal position of the ball after 0.5 s.

19. Practice Problem #2 A ball is thrown horizontally at 22.2 m/s from the roof of a building. It lands 36 m away from the building. How tall is the building?

20. Practice Problem #3 A diver running at 1.6 m/s dives out horizontally from the edge of a vertical cliff and reaches the water below 3.0 s later. How high was the cliff and how far from the base did the diver hit the water?

21. Section C: General Launch Angle • Corresponding Book Sections: • 4.4 • PA Assessment Anchors • S11.C.3.1

22. General Launch Angle • A particle launched at some angle above the horizontal • These are considerably more difficult than the zero-launch angle problem

23. What is different? • We need to break the initial velocity into x and y directions. • We may need to use the quadratic equation to solve for time  vox = vocosθ voy = vosinθ

24. Quadratic Equation • Use when solving for time in 2nd equation:

25. Practice Problem #1 • Refer to Easi-Teach file