Ch 03 Kinematics in Two Dimensions

Ch 03 Kinematics in Two Dimensions - Chapter 3 Kinematics...

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Chapter 3 y t i P r Q Path of particle r i t f r f O x A particle moving in the xy plane is located with the position vector r drawn from the origin to the particle . The displacement of the particle as it moves from P to Q in the time interval t = t f t i is equal to the vector r = r f r i . Kinematics in Two Dimensions The displacement, velocity, and acceleration The displacement vector r r f r i We define the “ Average velocity ” of the particle during the time interval t as the ratio of the displacement to that time interval: v ( r f r i ) ( t f t i ) = r t The “ instantaneous velocity ” , v , is defined as the limit of the average velocity , r / t , as t 0 : v im ( r / t) = d r / dt t 0 The magnitude of the instantaneous velocity ” vector is called the speed
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y P v i v v f r i Q v i r f v f O x The average acceleration vector a for a particle moving from P to Q is in the direction of the change in velocity , v = v f v i The “ average acceleration ” , a , is defined as the the ratio of the change in the instantaneous velocity vector, v , to the elapsed time , t : a ( v f v i ) / ( t f t i ) = v / t The instantaneous acceleration ” , a , is defined as the limiting value of the ratio v / t as t approaches zero : a im ( v / t ) = d v / dt t 0 It is important to recognize that various changes may constitute acceleration of a particle : (1) The magnitude of the velocity vector (speed) may change with time . (2) Only the direction of the velocity vector may change with time as its magnitude (speed) remains constant . (3) Both the magnitude and direction of the velocity vector may change .
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v = v o + a t velocity vector as a function of time . r = r o + v o t+ 1/2 a t 2 Position vector as a function of time . For simplicity take r o =0 . Thus in component form : v x = v xo + a x t v = v o + a t v y = v yo + a y t x = v xo t + 1/2 a x t 2 r = v o t + 1/2 a t 2 y = v yo t + 1/2 a y t 2 Because a is assumed constant , its components a x and a y are also constants. Thus, we can apply the equations of kinematics to the x and y components of the velocity and position vectors .
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