# 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 = constant

v = vo + a t

r = ro + vo t + 1/2 a t2    Displacement, et al Projectile Motion Return to ToC, Vectors and 2D Motion