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Path

Summary: Three Coordinates (Tool). Velocity. Acceleration. Reference Frame. Reference Frame. Path. Path. x. x. r. y. Observer’s measuring tool. r. y. O. Observer. Observer. (x,y) coord. (n,t) coord velocity meter. (r, q ) coord. r. q. Choice of Coordinates.

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Path

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

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

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

  4. A B O 2/8 Relative Motion (Translating axises) • Sometimes it is convenient to describe motions of a particle “relative” to a moving “reference frame” (reference observer B) • If motions of the reference axis is known, then “absolute motion” of the particle can also be found. • A = a particle to be studied • B = a “(moving) observer” Reference frame O Reference frame B • Motions of A measured by the observer at B is called the “relative motions of A with respect to B” • 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. frame work O is considered as fixed (non-moving)

  5. y A 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 x B O Relative position • If the observer at B use the x-y ** coordinate system to describe the position vector of A we have Y • Here we will consider only the case wherethe x-y axis is not rotating (translate only) X **other coordinates systems can be used; e.g. n-t.

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

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

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

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

  10. v y y x x vA/B vA vB Velocity Diagram aA aB aA/B Acceleration Diagram

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

  12. 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 vA of the skier and the value of 10 m o 30 Relative Motion: (Cicular Motion) Consider at point A and B as r- coordinate system 30 A A o o B B 30 D ? O.K. ? M Point: Most 2 unknowns can be solved with 1 vector (2D) equation.

  13. 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.

  14. t n (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. r  v a 

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