If an object moves in a curved path, it is said to be in curvilinear
motion. It is useful to use vectors to describe the motion of a curved
path.

Position

Position Vector

To define the position of a point P, select a fixed coordinate system with an origin O. The position of the point P is defined by a position vector, r, connecting O to P.

If P is in motion relative to O, then the position vector will change with time,

Usually, instantaneous velocity is needed, not average velocity. To
determine the instantaneous velocity, take the average velocity over
smaller and
smaller
increments
of time,

To determine the instantaneous acceleration, take the average acceleration
over smaller and smaller increments of time, giving

a = dv/dt

a = dv/dt

Rectangular Coordinates

Rectangular Coordinate System

Let the origin O reside at the origin of the Cartesian axis. Resolve
the position vector r(t) into rectangular x, y, and z components,

r = xi + yj + zk

Velocity

Differentiate to find velocity, giving

Acceleration

To find the acceleration, take a derivative of the velocity, which gives,

Rectangular motion is useful if a_{x} depends only on t, x, and/or v_{x}
(likewise for a_{y} and a_{z}).

Note that the equations describing the motion in each coordinate direction
are identical to those of linear motion, as denoted in the Position,
Velocity, Acceleration section.

2-D Projectile Problems

Most problems involve projectiles with constant gravity acceleration in only the y direction. In this special case, the rectilinear motion equations become (assumes up is positive y),

Practice Homework and Test problems now available in the 'Eng Dynamics' mobile app
Includes over 400 problems with complete detailed solutions.
Available now at the
Google Play Store and
Apple App Store.