Unit Kinematics Module Describing Motion in Two and Three Dimensions page 1 of 2 The Acceleration Vector v d v The vector formulation of instantaneous acceleration is a lim t 0 t dt The acceleration vector can be decomposed into a component a parallel to the velocity vector which describes changes in the magnitude of v and a component a perpendicular to the velocity vector which describes changes in the direction of v Two dimensional motion can be described using x and y component forms of the equations of kinematics The curve shows the path of a particle moving in two dimensions The vector emanating from the origin is the position or displacement vector and it describes the location of the particle at a certain time The other vector is the velocity vector and it describes the change in position of a particle per unit time It is tangent to the position curve Speed it the length of the velocity vector In two or three dimensions average acceleration is v The vector that expressed as vector a t represents the change in velocity v begins at the tip of the initial velocity vector v i and ends at the tip of the final velocity vector vf when v i and vf are placed tail to tail The acceleration vector is in the same direction as v however its magnitude may be different The instantaneous acceleration vector is found by taking the limit of the average acceleration vector as v d v t tends to zero a lim t 0 t dt At a certain point in time a particle s motion is described by its velocity and acceleration The acceleration vector can be decomposed into components that are parallel a and perpendicular a to the velocity vector If a is in the opposite direction from v the particle s speed is decreasing If a is in the same direction as v the particle s speed is increasing The component of acceleration perpendicular to v indicates changes in the direction of v www thinkwell com info thinkwell com Copyright 2001 Thinkwell Corp All Rights Reserved 1813 doc rev 03 27 2001 Unit Kinematics Module Describing Motion in Two and Three Dimensions page 2 of 2 The Acceleration Vector A useful way to think about the components of a vector is as the shadow or projection of a particle s motion in the x direction illustration on left and ydirection illustration on right You can easily think about motion on the x or y axis because it is onedimensional motion and you have already derived equations of kinematics in one dimension For example you can derive the first equation of kinematics from the definition of the average acceleration vector The vector equation vf v i a t can be decomposed into its xcomponents v fx v ix ax t and its ycomponents v fy v iy ay t The component forms of the vector equation simply describe onedimensional motion www thinkwell com info thinkwell com Copyright 2001 Thinkwell Corp All Rights Reserved 1813 doc rev 03 27 2001
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