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 = v_x i + v_y j

a = a_x i + a_y j

For 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 is

v_x = v_xo + a_x t

And we already know how the velocity component v_y depends upon a_y; that is

v_y = v_yo + a_y t

So 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 t i+ a_y t j )

v = ( v_xo i + v_yo j ) + ( a_x i + a_y j ) t

v = v_o + a t

Remember, v, v_o, and a are all vectors. As always, this vector equation is elegant shorthand notation for the two scalar equations,

v_x = v_xo + a_x t

and

v_y = v_yo + a_y 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 = x_o + v_xo t + ¹/₂ a_x t²

and

y = y_o + v_yo t + ¹/₂ a_y t²

We can combine these two scalar component equations, using the ideas and techniques of vector addition,

r = x i + y j

r = ( x_o + v_xo t + ¹/₂ a_x t² ) i + ( y_o + v_yo t + ¹/₂ a_y t² ) j

r = x_o i + v_xo t i + ¹/₂ a_x t² i + y_o j + v_yo t j + ¹/₂ a_y t² j

r = ( x_o i + y_o j ) + ( v_xo t i + v_yo t j ) + ( ¹/₂ a_x t² i + ¹/₂ a_y t² j )

r = ( x_o i + y_o j ) + ( v_xo i + v_yo j ) t + ¹/₂ ( a_x i + a_y j ) t²

r = r_o + v_o t + ¹/₂ a t²

As always, this vector equation is elegant shorthand notation for the two scalar equations,

x = x_o + v_xo t + ¹/₂ a_x t²

and

y = y_o + v_yo t + ¹/₂ a_y t²

---

a = constant

v = v_o + a t

r = r_o + v_o t + ¹/₂ a t²

Displacement, et al			Projectile Motion
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(c) 2002, Doug Davis; all rights reserved