1 O Path Reference Frame (x,y) coord r  (r,  ) coord x yr Path Reference Frame x yr (n,t) coord velocity meter Summary: Three Coordinates (Tool) Velocity Acceleration Observer Observer’s measuring tool Observer

2 O Path Reference Frame (x,y) coord r  (r,  ) coord x yr Path Reference Frame x yr (n,t) coord velocity meter Choice of Coordinates Velocity Acceleration Observer Observer’s measuring tool Observer

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4 Path (x,y) coord r  (r,  ) coord (n,t) coord velocity meter Translating Observer Two observers (moving and not moving) see the particle moving the same way? Observer O (non-moving) Observer’s Measuring tool Observer (non-rotating) Two observers (rotating and non rotating) see the particle moving the same way? Observer B (moving) Rotating No! “Translating-only Frame” will be studied today Which observer sees the “true” velocity? both! It’s matter of viewpoint. “Rotating axis” will be studied later. Point: if O understand B’s motion, he can describe the velocity which B sees. This particle path, depends on specific observer’s viewpoint “relative” “absolute” A “translating” “rotating”

5 2/8 Relative Motion (Translating axises)  A = a particle to be studied A Reference frame O frame work O is considered as fixed (non-moving)  If motions of the reference axis is known, then “absolute motion” of the particle can also be found. O  Motions of A measured using framework O is called the “absolute motions”  For most engineering problems, O attached to the earth surface may be assumed “fixed”; i.e. non-moving.  Sometimes it is convenient to describe motions of a particle “relative” to a moving “reference frame” (reference observer B) B Reference frame B  B = a “(moving) observer”  Motions of A measured by the observer at B is called the “relative motions of A with respect to B”

6 Relative position  If the observer at B use the x-y ** coordinate system to describe the position vector of A we have where = position vector of A relative to B (or with respect to B), and are the unit vectors along x and y axes (x, y) is the coordinate of A measured in x-y frame ** other coordinates systems can be used; e.g. n-t. B A X Y x y O  Here we will consider only the case where the x-y axis is not rotating (translate only)

7 B A X Y x y O  x-y frame is not rotating (translate only) Relative Motion (Translating Only) Direction of frame’s unit vectors do not change 0 0 Notation using when B is a translating frame. Note: Any 3 coords can be applied to Both 2 frames.

8 Understanding the equation Translation-only Frame! Path Observer O Observer B This particle path, depends on specific observer’s viewpoint A reference framework O O B reference frame work B A Observer O Observer B (translation-only Relative velocity with O) This is an equation of adding vectors of different viewpoint (world) !!! O & B has a “relative” translation-only motion

9 The passenger aircraft B is flying with a linear motion to theeast with velocity v B = 800 km/h. A jet is traveling south with velocity v A = 1200 km/h. What velocity does A appear to a passenger in B ? Solution x y

10 Translational-only relative velocity You can find v and a of B

11 vAvA vBvB v A/B Velocity Diagram x y aAaA aBaB a A/B Acceleration Diagram x y v

? B Yes No O Is observer B a translating-only observer relative with O

13 To increase his speed, the water skier A cuts across the wake of the tow boat B, which has velocity of 60 km/h. At the instant when  = 30°, the actual path of the skier makes an angle  = 50° with the tow rope. For this position determine the velocity v A of the skier and the value of Relative Motion: (Cicular Motion)  30 m10 A B  30 A B  D M ? ? O.K. Point: Most 2 unknowns can be solved with 1 vector (2D) equation. Consider at point A and B as r-  coordinate system

14 2/206 A skydriver B has reached a terminal speed. The airplane has the constant speed and is just beginning to follow the circular path shown of curvature radius = 2000 m Determine (a) the vel. and acc. of the airplane relative to skydriver. (b) the time rate of change of the speed of the airplane and the radius of curvature of its path, both observed by the nonrotating skydriver.

15 (b) the time rate of change of the speed of the airplane and the radius of curvature of its path, both observed by the nonrotating skydriver. n t

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17 v a  r 