Phy 211 General Physics I Chapter 4 Motion in 2 3 Dimensions Lecture Notes 2 Dimensional Motion A motion that is not along a straight line is a twodimensional motion The position displacement velocity and acceleration vectors are necessarily not on the same directions the rules for vector addition and subtraction apply The instantaneous velocity vector v is always tangent to the trajectory but the average velocity vavg is always in the direction of R The instantaneous acceleration vector a is always tangent to the slope of the velocity at each instant in time but the average acceleration aavg is always in the direction of v RA t0 v t RB t1 Example of 2 D Motion Displacement Velocity Accelerations When considering motion problems in 2 D the definitions for the motion vectors described in Chapter 2 still apply However it is useful to break problems into two 1 D problems usually Horizontal x Vertical y r r r Displacement Dr r ro x xo i y yo j r r r r dr drx dry v i j Velocity dt dt dt r r r r dv dv x dv y i j Acceleration a dt dt dt Use basic trigonometry relations to obtain vertical horizontal components for all vectors Equations of Kinematics in 2 D When to 0 and a is constant Horizontal Vertical vx vox axt x xo vox vx t vy voy ayt y yo voy vy t x xo voxt axt2 y yo voyt ayt2 vx2 vox2 2ax x vy2 voy2 2ay y Remember when to 0 replace t with t in the above equations Projectile Motion is the classic 2 D motion problem Vertical motion is treated the same as free fall ay g 9 8 m s2 downward Horizontal motion is independent of vertical motion but connected by time but no acceleration vector in horizontal direction ax 0 m s2 As with Free Fall Motion air resistance is neglected Projectile Motion Notes Projectile Motion Notes cont Uniform Circular Motion r v object travels in a circular path radius of motion r is constant speed v is constant velocity changes as object continually changes direction since v is changing v is not but direction is the object must be accelerated r r r dvx dvy ac ax ay dt i dt j the direction of the acceleration vector is always toward the center of the circular motion is referred to as the centripetal acceleration 2 r v the magnitude of centripetal acceleration ac ac r Derivation of Centrifugal Acceleration Consider an object in uniform circular motion where r constant radius of travel v constant speed of object r r dv The magnitude of the centripetal acceleration ac dt can be obtained by applying the Pythagorean theorem r ac a2x a2y The components of the velocity vector continuous change as the objects travels around the circle and are related to the angle 2 2 r d y d x d d ac vsin i vcos j v v dt dt dt r dt r 2 r v dy ac r dt 2 v dx r dt 2 2 r v dy dx ac r dt dt r v2 v 2 2 ac vy v x r r y v r x y x
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