PHY183-Lecture08 - M o t io n in 1 D im e n s io n ! First...

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Unformatted text preview: M o t io n in 1 D im e n s io n ! First case (last week): simple linear motion • Special case 1: motion with 0 acceleration • Special case 2: motion with constant acceleration • 2a: Free fall with a=-g Physics for Scientists & Engineers 1 Fall Semester 2006 Lecture 8 ! Other cases for later: 1) Circular motion • Motion in a 2-dimensional plane • 2D motion reducible to 1D motion by introducing new set of coordinates in the plane (polar coordinates). 2) Oscillations of an object attached to a spring • Motion due to force that is proportional to displacement • = “Restoring force” September 8, 2006 Physics for Scientists&Engineers 1 1 September 8, 2006 Physics for Scientists&Engineers 1 2 C o o r d in a t e S y s t e m ! In general, we use a 3-dimensional Cartesian coordinate system, with x-, y-, and z-axis ! Coordinate system has to be • Orthogonal: 90° angles between each pair of axes • Right-handed W o r k in g in C o m p o n e n t s ˆ ˆ ˆ ! Position vector r = ( x, y, z ) = xx + yy + zz ˆ ˆ ˆ ! Velocity vector v = ( vx , v y , vz ) = vx x + v y y + vz z vx = ! dx dy dz vy = vz = dt dt dt ! ! ˆ ˆ ˆ ! Acceleration vector a = ( ax , a y , az ) = ax x + a y y + az z ax = dv y dvx dv ay = az = z dt dt dt September 8, 2006 Physics for Scientists&Engineers 1 3 September 8, 2006 Physics for Scientists&Engineers 1 4 ! Special 3-D motion: the horizontal projection (in the x-yplane) is a straight line <=> motion in a plane ! We can introduce a new coordinate system such that x-axis of that new coordinate system lies on horizontal projection of the trajectory and y-axis points straight up ! Reduced 3d motion to effective 2d motion ! Example: gravity is the only force, acting in -z direction September 8, 2006 Physics for Scientists&Engineers 1 5 P r o je c t ile M o t io n P r o je c t ile M o t io n 2 ! ˆ ˆ ! Position vector r = ( x, y ) = xx + yy ! Velocity vector ! ˆ ˆ v = ( vx , v y ) = vx x + v y y ! ˆ ! Acceleration vector a = (0, ! g ) = ! gy ! 2 different types of motion along the two directions: • F r e e fa ll fo r v e r tic a l d ir e c tio n ( y - a x is ) • M o tio n w ith c o n s ta n t v e lo c ity ( = z e r o a c c e le r a tio n ) in h o r iz o n ta l d ir e c tio n ( x - a x is ) September 8, 2006 Physics for Scientists&Engineers 1 6 Id e a l P r o je c t ile M o t io n ! “Ideal” refers to absence of any wind resistance or other friction effect ! Only force acting: gravity In d e p e n d e n c e o f x - a n d y -m o t io n ! In ideal projectile motion, the motion in x- and ycomponents are independent of each other ! Not a trivial statement • Needs to be shown experimentally • Otherwise our mathematical description will have to be modified Photo/Illustration: Chris Hill shooting free throw during 2003 “BasketBowl” ! Only true for ideal projectile motion, in the case that we can neglect wind resistance • With wind resistance, the drag force is proportional to v2 and thus make the motions in x- and y-directions dependent on each other. September 8, 2006 Physics for Scientists&Engineers 1 7 September 8, 2006 Physics for Scientists&Engineers 1 8 E q u a t io n fo r Id e a l P r o je c t ile M o t io n ! Acceleration (to be explained later) ! Acceleration along each axis: F lig h t P a t h ! L e t s d e s c r i b e t h e s h a p e o f t h e t r a j e c t o r y i n t h e x y -p la n e !! a=g !ax (t ) = 0 " # "a y (t ) = $ g % ! Procedure: solve the equation x(t ) = x0 + vx 0t for the time: x(t ) ! # " y ( x) y (t ) $ t = ( x(t ) ! x0 ) / vx 0 ! N o w in se r t t h is in t o y ( t ) : ! Horizontal motion: constant velocity vx (t ) = vx 0 x(t ) = x0 + vx 0t y (t ) = y0 + v y 0t " gt # y = y0 + v y 0 1 2 2 ! Vertical motion: free fall v y (t ) = v y 0 ! gt y (t ) = y0 + v y 0t ! 1 gt 2 2 x " x0 1 ! x " x0 $ " 2 g% & vx 0 ' vx 0 ( 2 vx ! gx 2 " ! v gx " g y = $ y0 # y 0 0 # 20 % + $ y 0 + 20 % x # 2 x 2 vx 0 2vx 0 ' & vx 0 2vx 0 ' 2vx 0 & ! y = f(x2,x1,x0) => functional shape is a parabola 9 September 8, 2006 Physics for Scientists&Engineers 1 10 ! Use notation convention: September 8, 2006 vx 0 ! vx (t = 0); v y 0 ! v y (t = 0) Physics for Scientists&Engineers 1 F lig h t P a t h ( 2 ) ! Simplify: move origin so that x0=0 2 vy 0 x0 gx0 % " vy 0 gx0 % " g y = $ y0 ! ! 2 ' +$ + 2 x ! 2 x2 vx 0 2 vx 0 & # vx 0 2 vx 0 ' 2 vx 0 # & F lig h t P a t h ( 3 ) ! Visual confirmation y = y0 + vy 0 vx 0 x! g2 x 2 2 vx 0 ! Introduce 2 2 v0 = vx 0 + vy 0 " vy 0 % ! 0 = arctan $ ' # vx 0 & ! Result: y = y0 + x tan ! 0 " g x2 2 v cos 2 ! 0 2 0 11 September 8, 2006 Water fountain inside DTW airport terminal September 8, 2006 Physics for Scientists&Engineers 1 Physics for Scientists&Engineers 1 12 ...
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