Two-Dimensional Motion
With Constant Acceleration
We can write the vector displacement r in terms of its x- and y-coordinates,
r = x i + y j And, likewise for the vector velocity v and the vector acceleration a,
v = vx i + vy j a = ax i + ay j
For constant acceleration, this means ax = const and ay = const. We already know how the velocity component vx depends upon ax; that is
vx = vxo + ax t And we already know how the velocity component vy depends upon ay; that is
vy = vyo + ay t So we can write
v = vx i + vy j v = (vxo + ax t) i + (vyo + ay t) j
v = ( vxo i + vyo j ) + ( ax t i+ ay t j )
v = ( vxo i + vyo j ) + ( ax i + ay j ) t
v = vo + a t
Remember, v, vo, and a are all vectors. As always, this vector equation is elegant shorthand notation for the two scalar equations,
vx = vxo + ax t and
vy = vyo + ay t For constant acceleration, we already know how to relate the displacement to the initial displacement, initial velocity, and acceleration. That is, we already know
x = xo + vxo t + 1/2 ax t2 and
y = yo + vyo t + 1/2 ay t2 We can combine these two scalar component equations, using the ideas and techniques of vector addition,
r = x i + y j r = ( xo + vxo t + 1/2 ax t2 ) i + ( yo + vyo t + 1/2 ay t2 ) j
r = xo i + vxo t i + 1/2 ax t2 i + yo j + vyo t j + 1/2 ay t2 j
r = ( xo i + yo j ) + ( vxo t i + vyo t j ) + ( 1/2 ax t2 i + 1/2 ay t2 j )
r = ( xo i + yo j ) + ( vxo i + vyo j ) t + 1/2 ( ax i + ay j ) t2
r = ro + vo t + 1/2 a t2
As always, this vector equation is elegant shorthand notation for the two scalar equations,
x = xo + vxo t + 1/2 ax t2 and
y = yo + vyo t + 1/2 ay t2
a = constantv = vo + a t
r = ro + vo t + 1/2 a t2
ToC, 2-D Kinematics Projectile Motion Return to ToC, Vectors and 2D Motion (c) 2005, Doug Davis; all rights reserved