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HR4 - Chapter 4 Motion in Two and Three Dimensions In this...

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Unformatted text preview: Chapter 4 Motion in Two and Three Dimensions In this chapter we will continue to study the motion of objects without the restriction we put in chapter 2 to move along a straight line. Instead 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 r Example : 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) P 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 the same approach as in 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 2 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 avg r r r r v ∆ ∆ ∆...
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HR4 - Chapter 4 Motion in Two and Three Dimensions In this...

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