Review 1D motion. Other 1D equations (in table 2-1) are restatements of these

Transcription

1 Review 1D motion Constant acceleration Position (x) and velocity (v) as a function of time Note when clock starts (t= ), the position and velocity are x and v v = v + at 1 x = x + vt + at Other 1D equations (in table -1) are restatements of these slide 1

2 One of HW Problems 1 x = x + vt + at Graph given is an expression of one of the derived equations. In this case, we are given every position (x) for every t for <t<s In particular, three black points satisfy equation Here, what we don t know are x, v, and a. Three relationships among x, v, and a are enough to find them! slide

3 Another HW problem Initially, train going faster than locomotive Assume constant deceleration for train What is this -- so they don t collide? v = v + at 1 x = x + vt + at Criteria (limiting case): When they meet 1. Just touch. Have same speed equations unknowns slide 3

4 Position, velocity, acceleration vectors review r v 1 v dt Position vector from origin to present location Velocity vector is along trajectory Acceleration vector can be any direction a v = dr dv = = dt dr dt slide 4

5 Scalar Product of Vectors S a b = ab = = ( acos φ) ( b) ( φ ) ( a) bcos The product of the common components Note limiting cases f = and f = 9 cosφ Example of use: discussion of WORK (later) slide 5

6 Vector Product - 1 Needs 3D for description Different product of two vectors (from scalar pr.) result a vector Note limiting cases f = and f = 9 Examples of use: torque and magnetism c a b c c = ab sinφ c a and c b slide 6

7 Vector Product - c a b = ( b c c = ab sinφ c a and c b a) Two methods to remember direction of vector product Right hand rule Direction that a right hand screw advances when rotating a into b slide 7

8 Reminder: Real World Motion (on Earth) If all other forces (friction, air drag, ) are small and can be neglected, then horizontal motion has zero acceleration vertical motion has a universal value of acceleration: GOT IT RIGHT! a=g=9.8 m/s =3 ft/s DOWN! slide 8

9 Juggler and Balls one ball-parabolic path.mov 3balls-parabolic.MOV horizontal and vertical acceleration when ball is in the air (in FREE FALL) When ball is touched, acceleration different Note trajectory is parabola slide 9

10 D motion near Earth s surface a y (x,y ) v x Critical points: Relative to origin, only two vectors: a = gjˆ and v So motion in a plane: D All bodies experience acceleration = -g in y dir independent of the horizontal motion (so long as friction, drag, etc, can be neglected) Demo: shoot the monkey! 1 y = y + vyt gt slide 1

11 Trajectory: Each dimension has const acceleration v = v + at Horizontal: a x = So, it terms of initial x-component velocity and position 1 x = x + vt + at Vertical: a y =-g So, in terms of initial y-component velocity and position y a (x,y ) v x v x = v x x = x + v t x v = v gt y y 1 y = y + vyt gt slide 11

12 Trajectory equation a = gjˆ x-comp of v is fixed y-comp of v is getting more negative at a fixed rate (g) balls-traj eq.mov 1 g y = (tan θ )( x) ( x) ( vcos θ) Choose coord system with x = y = Movie showing parabolic trajectory for projectile is ALWAYS the case while ball is in the air (FREE FALL). Specific shape of parabola depends on INITIAL CONDITIONS. slide 1

13 Work on the consequences 1 g y = (tan θ )( x) ( x) ( vcos θ) Range, Sample problems: 4-6, 7, 8 + HW probs Same relation: different initial conditions Note double valued in angle v Range = sin( θ) g slide 13

14 Motion in a Circle at Constant Speed circ motion- v chg.mov Graphical argument for the sign of the centripetal acceleration movie a = dv dt slide 14

15 v π r r θ Circles and Circular Motion r θ v r are changing v = v = constant r = r = constant vt = Object moving at constant speed v in a circle, rad r Velocity vector always perpendicular to position vector Assume time for one circumference is T a v slide 15

16 Finish for the Day Make sure to understand chapter 4 We have discussed Trajectory motion Circular motion Next time Relative motion (complete chapter 4) Forces (chapter 5) Reminder First homework due on Monday (ch, 3, 4) bring to class and deposit in indicated box before class Those unable to make lecture on Monday, make sure it gets to me then or beforehand (under office door) or my mailbox (#1) across from physics dept. office on 7 th floor Pupin slide 16