CH03 - Summary (CH1 and CH2) Vectors decomposition Vector...

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Summary (CH1 and CH2) Vectors decomposition Vector addition 1d motion (constant acceleration) V = V x ˆ i + V y ˆ j = | V | cos θ ˆ i + | V | sin ˆ j V tot = ( V 1 x + V 2 x ) ˆ i + ( V 1 y + V 2 y ) ˆ j tg = V 1 y + V 2 y V 1 x + V 2 x | V tot | = V 1 x + V 2 x ) 2 + ( V 1 y + V 2 y ) 2 v avg = Δ x/ Δ t = (x-x 0 )/(t-t 0 ) a avg = Δ v/ Δ t = (v x -v 0x )/(t-t 0 ) v x = v 0 x + a x t x = x 0 + v 0 x t + 1 2 a x t 2 v x 2 = v 0 x 2 + 2 a x ( x x 0 ) x x 0 = 1 2 ( v x + v x 0 ) t Average velocity/acceleration Instantaneous velocity/acceleration v x = dx dt a x = dv x dt
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Goals for Chapter 3: 2D or 3D motions position, velocity, and acceleration vectors (2D/3D) projectile motion Intial velocity + Gravitational acceleration circular motion Uniform circular motion Non-uniform circular motion relative velocity
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Opening question When the race car goes around a curve, is it accelerating? If it is, in what directions? If I drop a pen, does it hit the ground at the same time as one throw horizontally?
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Position relative to the origin—Figure 3.1 An overall position relative to the origin can have components in x , y , and z dimensions. Straight line motion is the exception, not the norm The path for a particle is generally a curve.
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Displacement and Average velocity (2d) vectors The average velocity between two points will have the same direction as the displacement. Note: In straight line motion, Velocity is either positive, or negative In 2d-3d motion, displacement and velocity is defined by its components in all 2(3) directions v = v x ˆ i + v y ˆ j Δ r = ( Δ x ) ˆ i + ( Δ y ) ˆ j Compare graphically! Which is head to tail? Which is tail to tail? r 2 + r 1 and r 2 r 1
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Components of velocity(2D) A velocity in the xy -plane can be decomposed into separate x and y components. 1d 2d v avg = Δ x Δ t v x = dx dt v avg = r 2 r 1 t 2 t 1 = Δ r Δ t v = Δ t 0 lim Δ r Δ t = d r dt v x = dx dt v y = dy dt v = dx dt ˆ i + dy dt ˆ j
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Short Quiz If the position components are know as a function of time x(t), y(t) (and z(t) in 3d) are given Then what do we know at any given time: A. Position vector B. Average velocity C. Instantaneous velocity D. Average acceleration E. Instantaneous acceleration F. All of the above
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Instantaneous velocity—Example The figure blow shows velocity and acceleration as time passes. A rover’s x,y-coordinates vary with time: What is the rover’s instantaneous velocity vector? What the magnitude and direction of this vector at t=2.0s? x = 2.0 m (0.25 m / s 2 ) t 2 y = (1.0 m / s ) t + (0.25 m / s 3 ) t 3
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