Physics · Ch 3 — Motion in a Plane

Motion in a Plane with Uniform Velocity and Uniform Acceleration

3.11

The equations of motion developed for a straight line extend directly to two dimensions once

they are written as vector equations. For a particle with constant acceleration

$\vec a$ , initial velocity $\vec v_0$ at position $\vec r_0$ (all measured at $t = 0$ ), the

velocity and position at any later time $t$ are

\vec v = \vec v_0 + \vec a\,t, \qquad \vec r = \vec r_0 + \vec v_0\,t + \tfrac{1}{2}\vec a\,t^2.

These are exactly the same equations used for straight-line motion, except that $\vec v_0$ ,

$\vec a$ , and $\vec r$ are now genuine two-dimensional vectors rather than signed numbers.

Principle of independence of motion. The real power of the vector form is that each of

these vector equations is equivalent to two separate, independent scalar equations — one

for the x-direction and one for the y-direction:

v_x = v_{0x} + a_x t, \qquad x = x_0 + v_{0x}t + \tfrac12 a_x t^2,

v_y = v_{0y} + a_y t, \qquad y = y_0 + v_{0y}t + \tfrac12 a_y t^2.

Motion along the x-axis and motion along the y-axis proceed completely independently of each

other, each governed by the familiar one-dimensional equations with its own initial velocity

and its own acceleration component. This is the principle of independence of motion, and

it is the single most important idea used to solve every two-dimensional motion problem in

this chapter, from projectile motion to uniform circular motion.

Uniform velocity is the special case $\vec a = 0$ : the particle moves along a straight

line at constant velocity, and $\vec r = \vec r_0 + \vec v_0\,t$ .

Uniform acceleration in general — with $\vec a$ constant but not necessarily parallel to …