Chapter 3 Kinematics in More Dimensions; Vectors

Vectors & Scalars 2 Magnitude Direction Vector Only magnitude Scalar Displacement, velocity, acceleration Force, momentum, torque … Distance traveled, speed, mass, time … Such as

3 Vector algebra a a=|a |x axax ayay azaz y z o a Vector Diagram of vector Magnitude

4 Addition and Subtraction a b a+b a b a - b a b + =？=？ a c a+b+c b ba c =? - a b =？=？

5 Scalar product b a 

6 Vector product b a  where Right-hand rule

7 Position vector z O x y P(x,y,z) Motional equation ： or cancel t Path equation

8 Two kinds of equations Motional equation of a projectile motion cancel t If a particle move as Path equation? What motion if a = b? parabola ellipse

9 P1P1 P2P2 Displacement x y z O Distance traveled: ΔS Length of path A B When  t → 0, |Δr |   S 。

10 Vector kinematics Velocity Acceleration Speed Motions in components (superposition principle): 3 independent motions in each dimension

11 Concept of vector Consider uniform circular motion: Change in both magnitude and direction

12 Solving problems Example1: The position of a particle is given by a) Calculate: when s. b) When ? Solution: a) b) When,.

13 Example2: A particle is at rest in origin when t=0, and it has an acceleration as a) Where the particle is when t=2s? b) What is the path like? Solution: a) b) The path is a parabola.

14 Challenging question Question: The position of a particle is given by What is the distance traveled in 0~1s ? constant

Projectile motion 15 Horizontal: motion with constant velocity v x Vertical: falling object with velocity v y

16 Motional equation Path equation

17 Wrong strategy Example3: A boy hanging from a tree is straightly aimed by a water-balloon slingshot horizontally. He falls from the tree when the slingshot shots, hoping to avoid being hit. Can he escape? Solution: The boy and the water balloon have the same vertical motion: So he can not escape!

18 Shooting range Example4: A cannonball is been shooting to a hill, with initial velocity 150m/s at angle 60°to the horizontal, what is the range? v 60° A O 30° x y O Solution: y x maximum range?

19 Uniform circular motion Speed remains constant Angular coordinate  Angular velocity y x o  r s Angular acceleration Frequency & Period

20 Radial acceleration Direction of is tangential Acceleration of UCM oo p1p1 r p2p2    when, radial acceleration

21 Nonuniform circular motion Tangential acceleration How the magnitude of velocity changes along the line tangent to the path Centripetal/radial/normal acceleration How the direction of velocity changes toward the center of the circle

22 Properties of acceleration v a tan aRaR Perpendicular Angular quantities: Any curved path p For :

23 Moving in circle Example5: An object moving in r=2m circle satisfies  =2  +6  t -  t 2 (SI) a) Calculate a tan, a R when t=1s. b) When would it stop? Solution: a) b)

24 Dangerous car Example6: A car is running in a circle of r = 100m, it starts at t = 0s and a tan =3m/s 2, what is the magnitude of total acceleration at t = 10s ? Solution:

25 Homework An object moving in r =100m circle satisfies s =36t - 2t 3 (SI) Calculate a tan, a R and a when t=1s.

26 Relative motion r ps = r ps + r ss y x y o S S z z O r ps. p. p rssrss v ps = v ps + v ss a ps = a ps + a ss How to describe motions in different frames S’ moves in translational way to S Galilean transformations

27 Relative velocity v ps = v ps + v ss Velocities in different frames are connected by this vector addition equation. 1. Subscripts: first refers to the object second to the reference frame 2. Drawing diagrams will be useful. Notice:

28 Wind blow Example7: Wind blows 10km/h east to west, and river flows 10km/h from west to east, a boat moves on the river with relative speed 20km/h, 30°north by west. How about the direction of wind relative to the boat? x y o v BR =20 30° 30  Solution: 30°south by west

29 Shoot falling object Example8: A man is aimed by a gun, and he starts to fall down when the gun fires, can he escape? Solution: G M In reference frame of the Gun Addition of two motions In reference frame of the Man Motion with constant velocity

30 Summary of kinematics