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# Lect_05 - ACT Alice and Bob Lecture 5 2D and 3D motion...

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Lecture 5 2D and 3D motion ACT: Alice and Bob ACT: Alice and Bob Alice and Bob stand at the top of a cliff of height h . Both throw a ball with initial speed v 0 , Alice straight up and Bob straight down. The speed of the balls when they hit the ground are v A and v B , respectively. Which of the following is true? Alice Bob v 0 v 0 v A v B A) v A < v B B) v A = v B C) v A > v B y y 0 = h v 2 = v 0 2 – 2 gh same for both! + v 0 or – v 0 2D (and 3D) motion Now we need vectors to indicate position, velocity and acceleration, but the definitions we use in 1D are pretty much the same. Position: ( ) r t G x y ( 3 s) r t = G ( 1 s) r t = G trajectory Displacement: ( ) ( ) r r t t r t = + ∆ G G G ( ) final initial or r r r = G G G r Velocity Velocity = G G Average: r v t = = = = G G Instantaneous: , , x y z dy dr dx dz v v v v dt dt dt dt G is always tangent to the trajectory. v v G ( 1 s) v t = ( 3 s) v t = G

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Acceleration Acceleration • Average: • Instantaneous: = G G v a t = = = = G G , , y x z x y z dv dv dv dv a a a a dt dt dt dt ACT: Acceleration Shown below are the trajectory of a moving object and the snapshots taken every second. Which of the following is true about the components of the acceleration? y x 1s 2s 3s 4s A) a x = 0, a y > 0 B) a x > 0, a y > 0 C) a x < 0, a y = 0 Note: Both the speed and the direction of velocity are changing! y x 1s 2s 3s 4s v (1) (1) v (2) v (3) v (1) v (3) v The big new thing in 2D: changes in direction Change in speed; parallel to v Change in direction; perpendicular to v ˆ ˆ ( ) ˆ ˆ dv d vv dv dv v vv a v v dt dt dt dt = = = = + G G G An object can move at constant speed and still have a 0! This didn’t happen in 1D!! ( ) v t G ( ) v t t + ∆ G v G a G ( ) v t G t t t + ∆ ( ) v t t + ∆ G Graphically: Imagine an object moving along the following trajectory at constant speed. Take the positions at times t and
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