An object accelerates when it changes velocity. In two dimensions, velocity can change by changing magnitude (speed) or by changing direction. These are represented by acceleration components tangent to and perpendicular to an object’s trajectory.

《 LOOKING BACK Section 1.5 Finding acceleration vectors on a motion diagram

Projectile motion is two-dimensional free-fall motion under the influence of only gravity. Projectile motion follows a parabolic trajectory. It has uniform motion in the horizontal direction and $a_y=-g$ in the vertical direction.

《 LOOKING BACK Section 2.5 Free fall

Coordinate systems that move relative to each other are called reference frames. If object C has velocity $\vec{v}{}_{\text{CA}}$ relative to a reference frame A, and if A moves with velocity $\vec{v}{}_{\text{AB}}$ relative to another reference frame B, then the velocity of C in reference frame B is $\vec{v}{}_{\text{CB}}=\vec{v}{}_{\text{CA}}+\vec{v}{}_{\text{AB}}$.

An object moving in a circle (or rotating) has an angular displacement instead of a linear displacement. Circular motion is described by angular velocity $\omega$ (analogous to velocity $v_s$) and angular acceleration $\alpha$ (analogous to acceleration $a_s$). We’ll study both uniform and accelerated circular motion.

An object in circular motion is always changing direction. The acceleration of changing direction—called centripetal acceleration—points to the center of the circle. All circular motion has a centripetal acceleration. An object also has a tangential acceleration if it is changing speed.

Linear motion allowed us to introduce the concepts of motion, but most real motion takes place in two or even three dimensions. Balls move along curved trajectories, cars turn corners, planets orbit the sun, and electrons spiral in the earth’s magnetic field. Where is two-dimensional motion used? Everywhere!