Transcription of Chapter 3 Motion in Two and Three Dimensions
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Chapter 3 Motion in Two and Three Dimensions
Chapter 3 Motion in Two and Three The Important PositionIn Three Dimensions , the location of a particle is specified by itslocation vector, r:r=xi+yj+zk( )If during a time interval tthe position vector of the particle changes fromr1tor2, thedisplacement rfor that time interval is r=r1 r2( )= (x2 x1)i+ (y2 y1)j+ (z2 z1)k( ) VelocityIf a particle moves through a displacement rin a time interval tthen its average velocityfor that interval isv= r t= x ti+ y tj+ z tk( )As before, a more interesting quantity is theinstantaneousvelocityv, which is the limitof the average velocity when we shrink the time interval tto zero. It is the time derivativeof the position vectorr:v=drdt( )=ddt(xi+yj+zk)( )=dxdti+dydtj+dzdtk( )can be written:v=vxi+vyj+vzk( )5152 Chapter 3. Motion IN TWO AND Three Dimensions wherevx=dxdtvy=dydtvz=dzdt( )The instantaneous velocityvof a particle is always tangent to the path of the AccelerationIf a particle s velocity changes by vin a time period t, the average accelerationaforthat period isa= v t= vx ti+ vy tj+ vz tk( )but a much more interesting quantity is the result of shrinking the period tto zero, whichgives us the instantaneous acceleration,a.
with v’s and a’s standing for the appropriate time derivatives, then we have the relations: rPA = rPB +rBA (3.23) vPA = vPB +vBA (3.24) For the purposes of doing physics, it is important to consider reference frames which move at constant velocity with respect to one another; for these cases, vBA = 0 and then we find
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