Chapter_4 - Chapter 4 Motion in Two and Three Dimensions In...

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Unformatted text preview: Chapter 4 Motion in Two and Three Dimensions In this chapter we will consider motion in a plane ( two dimensional motion) and motion in space ( three dimensional motion) The following vectors will be defined for two- and three- dimensional motion: Displacement Average and instantaneous velocity Average and instantaneous acceleration We will consider in detail projectile motion and uniform circular motion as examples of motion in two dimensions Finally we will consider relative motion, i.e. the transformation of velocities between two reference systems which move with respect to each other with constant velocity (4 -1) Position Vector The position vector of a particle is defined as a vector whose tail is at a reference point (usually the origin O) and its tip is at the particle at point P. The position vector in the f Example : r r igure is: ˆ ˆ ˆ r xi yj zk = + + r ( 29 ˆ ˆ ˆ 3 2 5 r i j k m = - + + r (4 -2) x y O (x,y) P (-3,2,5) t 2 t 1 Displacement Vector 1 2 For a particle that changes postion vector from to we define the as follows: r r r ∆ r r r displacement vector 2 1 r r r ∆ =- r r r 1 2 The position vectors and are written in terms of components as: r r r r 1 1 1 1 ˆ ˆ ˆ r x i y j z k = + + r 2 2 2 2 ˆ ˆ ˆ r x i y j z k = + + r ( 29 ( 29 ( 29 2 1 2 1 2 1 ˆ ˆ ˆ ˆ ˆ ˆ r x x i y y j z z k xi yj zk ∆ =- +- +- = ∆ + ∆ + ∆ r (4 -3) 2 1 x x x ∆ =- 2 1 y y y ∆ =- 2 1 z z z ∆ =- The displacement r can then be written as: ∆ r t t + Δt Average and Instantaneous Velocity Following chapter 2 we define the average velocity as: displacement average velocity = time interval ˆ ˆ ˆ ˆ ˆ ˆ avg r xi yj zk xi yj zk v t t t t t ∆ ∆ + ∆ + ∆ ∆ ∆ ∆ = = = + + ∆ ∆ ∆ ∆ ∆ r r We define as the instantaneous velocity (or more simply the velocity) as the limit: lim r dr v t dt t ∆ = = ∆ ∆ → r r r (4 - 4) t t + Δt 2 1 If we allow the time interval t to shrink to zero, the following things happen: Vector moves towards vector and The direction of the ratio (and thus )approaches t ∆ ∆ → ∆ ∆ 1. 2. avg r r r r v r r r r r the direction of the tangent to the path at position 1 → 3. avg v v r r ( 29 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x y z d dx dy dz v xi yj zk i j k v i v j v k dt dt dt dt = + + = + + = + + r (4 - 5) x dx v dt = y dy v dt = z dz v dt = The three velocity components are given by the equations: dr v dt = r r Average and Instantaneous Acceleration The average acceleration is defined as: change in velocity average acceleration = time interval 2 1 avg v v v a t t- ∆ = = ∆ ∆ r r r r We define as the instantaneous acceleration the limit:...
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This note was uploaded on 09/27/2010 for the course PHY 55555 taught by Professor Ia during the Spring '10 term at SUNY Buffalo.

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Chapter_4 - Chapter 4 Motion in Two and Three Dimensions In...

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