1. Chap. 3: Kinematics in Two or Three Dimensions: Vectors HW3: Chap. 2: Pb. 51, Pb. 63, Pb. 67; Chap 3:Pb.3,Pb.5, Pb.10, Pb.38, Pb.46 Due Wednesday, Sept. 16

2. Variable Acceleration; Integral Calculus Deriving the kinematic equations through integration: For constant acceleration,

3. Variable Acceleration; Integral Calculus Then: For constant acceleration,

4. Displacement from a graph of constant v x (t) Solve for displacement t1t1 t2t2 Displacement is the area between the v x (t) curve and the time axis t v x +∆x -∆x SIGN 0 t v x 0

5. Displacement from graphs of v(t) What to do with a squiggly v x (t)? o make ∆t so small that v x (t) does not change much t1t1 t2t2 Displacement is the area under v x (t) curve t v x ∆t Velocity does not need to be constant

6. Graphical Analysis and Numerical Integration Similarly, the velocity may be written as the area under the a-t curve. However, if the velocity or acceleration is not integrable, or is known only graphically, numerical integration may be used instead.

7. One Dimensional Kinematics https://www.youtube.com/watch?v=wNQzqCcTXR4&index=5&list=PLCF- Lie6gOOTx_CUIBUUXkhH2ezY8zcJB

8. Review Question A ball is thrown straight up into the air. Ignore air resistance. While the ball is in the air the acceleration A) increases B) is zero C) remains constant D) decreases on the way up and increases on the way down E) changes direction

9. Vector Addition: Graphical Vectors Scalars Magnitude andMagnitude only Direction Symbol r r Examples: Displacement Velocity acceleration Distance speed time Examples:

10. 2D Vectors Magnitude and direction are both required for a vector! How do I get to Washington from New York? Oh, it’s just 233 miles away.

11. Vector Addition: Graphical When we add vectors Order doesn’t matter We add vectors by drawing them “tip to tail ” A B start The resultant starts at the beginning of the first vector and ends at the end of the second vector

12. Vector Addition Question A B 1) 2) 3) Which graph shows the correct placement of vectors for + A B

13. Vector Addition Question A B 1) 2) 3) Which graph shows the correct resultant for + A B

14. Vector Subtraction: Graphical A B When you subtract vectors, you add the vector’s opposite. - = + - AB AB B - A C D A B -B A

15. Addition of Vectors—Graphical Methods The parallelogram method may also be used; here again the vectors must be tail-to-tip.

16. Multiplication of a Vector by a Scalar A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.

17. Vector Addition: Components If the components are perpendicular, they can be found using trigonometric functions.

18. Vector Addition: Components We don’t always carry around a ruler and a protractor, and our result isn’t always very precise even when we do. In this course we will use components to add vectors. However, you should still always draw the vector addition to help you visualize the situation. What are components here? A x y AxAx AyAy Parts of the vector that lie on the coordinate axes

19. Vector Addition: Components We add vectors by adding their x and y components because we can add things in a line A x y AxAx AyAy B y x BxBx ByBy C C AxAx ByBy BxBx AyAy A B

20. Vector Addition: Components We add vectors by adding their x and y components. C AxAx ByBy BxBx AyAy AxAx BxBx CxCx ByBy AyAy CyCy CxCx C CyCy

21. Vector Addition: Components Once we have the components of C, C x and C y, we can find the magnitude and direction of C. C CxCx CyCy South of East magnitude direction

22. Unit Vectors Unit vectors have magnitude 1. Using unit vectors, any vector can be written in terms of its components:

23. Adding Vectors by Components Example 3-2: Mail carrier’s displacement. A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office?

24. Vector Kinematics In two or three dimensions, the displacement is a vector:

25. Vector Kinematics As Δt and Δr become smaller and smaller, the average velocity approaches the instantaneous velocity.

26. Vector Kinematics The instantaneous acceleration is in the direction of Δ = 2 – 1, and is given by:

27. Vector Kinematics Using unit vectors,