## Two-Dimensional Motion

## With Constant Acceleration

We can write the vector displacement

rin terms of its x- and y-coordinates,r= xi+ yjAnd, likewise for the vector velocity

vand the vector accelerationa,v= v_{x}i+ v_{y}j

a= a_{x}i+ a_{y}jFor constant acceleration, this means a

_{x}= const and a_{y}= const. We already know how the velocity component v_{x}depends upon a_{x}; that isv _{x}= v_{xo}+ a_{x}tAnd we already know how the velocity component v

_{y}depends upon a_{y}; that isv _{y}= v_{yo}+ a_{y}tSo we can write

v= v_{x}i+ v_{y}j

v= (v_{xo}+ a_{x}t)i+ (v_{yo}+ a_{y}t)j

v= ( v_{xo}i+ v_{yo}j) + ( a_{x}ti+ a_{y}tj)

v= ( v_{xo}i+ v_{yo}j) + ( a_{x}i+ a_{y}j) t

v=v_{o}+atRemember,

v,v_{o}, andaare allvectors. As always, this vector equation is elegant shorthand notation for thetwoscalar equations,v _{x}= v_{xo}+ a_{x}tand

v _{y}= v_{yo}+ a_{y}tFor

constant acceleration, we already know how to relate thedisplacementto the initial displacement, initial velocity, and acceleration. That is, we already knowx = x _{o}+ v_{xo}t +^{1}/_{2}a_{x}t^{2}and

y = y _{o}+ v_{yo}t +^{1}/_{2}a_{y}t^{2}We can combine these two scalar component equations, using the ideas and techniques of vector addition,

r= xi+ yj

r =( x_{o}+ v_{xo}t +^{1}/_{2}a_{x}t^{2})i+ ( y_{o}+ v_{yo}t +^{1}/_{2}a_{y}t^{2})j

r =x_{o}i+ v_{xo}ti+^{1}/_{2}a_{x}t^{2}i+ y_{o }j+ v_{yo}tj+^{1}/_{2}a_{y}t^{2}j

r =( x_{o}i+ y_{o }j) + ( v_{xo}ti+ v_{yo}tj) + (^{1}/_{2}a_{x}t^{2}i+^{1}/_{2}a_{y}t^{2}j)

r =( x_{o}i+ y_{o }j) + ( v_{xo}i+ v_{yo}j) t +^{1}/_{2}( a_{x}i+ a_{y}j) t^{2}

r =r_{o}+v_{o}t +^{1}/_{2}at^{2}As always, this vector equation is elegant shorthand notation for the

twoscalar equations,x = x _{o}+ v_{xo}t +^{1}/_{2}a_{x}t^{2}and

y = y _{o}+ v_{yo}t +^{1}/_{2}a_{y}t^{2}

a=constant

v=v_{o}+at

r =r_{o}+v_{o}t +^{1}/_{2}at^{2}

Displacement, et alProjectile MotionReturn to ToC, Ch4, Two-Dimensional Motion(c) Doug Davis, 2001; all rights reserved