![sin cos
sin cos
ˆ ˆi j
x y
x y
x y
v r t v r t
v v t v v t
v v v
r
Analyzing the motion
cos sinp px r t y r t
p p
x y
dx dy
v v
dt dt
[remember, ]
v
v r
r
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Class9
- 1. Class 9 – Review of motion in 2D and 3D Chapter 4 - Monday September 13th •Exam instructions •Brief review •Example problems Reading: pages 58 thru 75 (chapter 4) in HRW Read and understand the sample problems •Exam tonight: 8:20-10:20pm WebAssign is NO LONGER FREE!! You must register
- 2. ˆ ˆ ˆi j kr x y z r 2 1 ˆ ˆ ˆi j kr r r x y z r r r ˆ ˆ ˆi j k dr dx dy dz v dt dt dt dt r r ˆ ˆ ˆi j kyx z dvdv dv dv a dt dt dt dt r r Two-dimensional kinematics Acceleration: Position: Displacement: Velocity:
- 3. 0 0 0 0 0 21 0 0 0 2 0 0 22 0 0 0 cos 4 21 cos sin 4 22 sin 4 23 sin 2 4 24 x y y x x v t v v y y v t gt v v gt v v g y y Projectile motion
- 4. Uniform circular motion •Although v does not change, the direction of the motion does, i.e. the velocity (a vector) changes. •Thus, there is an acceleration associated with the motion. •We call this a centripetal acceleration. 2 2 Acceleration: Period: 1 1 Frequency : ; 2 2 v r a T r v v v f f T r r •Since v does not change, the acceleration must be perpendicular to the velocity.
- 5. cos sin ˆ ˆi j p p p p x r y r r x y r Analyzing the motion 2 2 t f t t T Uniform motion: Note: 180 2 radianso cos sinp px r t y r t
- 6. sin cos sin cos ˆ ˆi j x y x y x y v r t v r t v v t v v t v v v r Analyzing the motion cos sinp px r t y r t p p x y dx dy v v dt dt [remember, ] v v r r
- 7. Acceleration ˆ ˆi j ˆ ˆ= i j yx x y dvdvdv a dt dt dt a a r r 2 2 2 22 2 cos sinx y v v a a a v r r sin cos cos sin x y x y v v t v v t a v t a v t recall : cos sinp px r t y r t 2 2 cos sinx y v v a t a t r r v r
- 8. Relative motion PA PB BAr r r r r r Position: PA PB BA PA PB BA PB R dr dr dr v v v v v dt dt dt r r r r r r r r Velocity: PA PB R PA PB dv dv dv a a dt dt dt r r r r r Acceleration (if frame B is “inertial”):