Particle Kinematics. Methods of Describing Motion

Particle kinematics describes the motion of a particle without considering the forces that cause it. The basic objective is to specify particle position as a function of time and determine its trajectory, velocity, and acceleration relative to a selected reference frame.

Reference frame and equations of motion

A motion description requires a reference body, an associated coordinate system, and a measure of time. The equations of motion must determine the particle position at any instant within the interval being studied.

Main methods of describing motion

1. Vector description. The particle position is specified by a position vector $\vec r=\vec r(t)$ measured from a chosen origin.
2. Cartesian-coordinate description. The functions $x=x(t)$, $y=y(t)$, and $z=z(t)$ specify the coordinates and together form the equations of motion.
3. Path-coordinate description. When the trajectory is known, the particle position is specified by a path coordinate $s=s(t)$ measured from a selected origin along the curve with an assigned positive direction.

Choose the description that makes the given data and required kinematic quantities easiest to use.

Vector description

The position vector $\vec r(t)$ extends from the coordinate origin to the moving particle. As time varies, the endpoints of $\vec r(t)$ trace the trajectory. In a Cartesian basis, $\vec r=x\vec i+y\vec j+z\vec k$.

Cartesian-coordinate description

The relations $x=x(t)$, $y=y(t)$, and $z=z(t)$ are the kinematic equations of motion. An equation of the trajectory can be obtained by eliminating time from these relations. Two coordinates are sufficient for planar motion.

Path-coordinate description

If the trajectory is already known, choose an origin $O_1$ on the curve, assign a positive direction, and specify $s=s(t)$. The sign of $s$ locates the particle relative to the path-coordinate origin, while the sign of $\dot s$ indicates its direction of motion along the trajectory.

Trajectory, distance traveled, and displacement

The trajectory is the geometric locus of successive particle positions. Distance traveled is the length accumulated along the trajectory and does not decrease as the particle moves. Displacement is the vector from the initial to the final position; its magnitude is generally not equal to the distance traveled.

Example

Suppose planar motion is given by $x=2t$ and $y=t^2$ in metres. Eliminating time with $t=x/2$ gives the trajectory $y=x^2/4$, a parabola. The parametric equations also specify where the particle is on that trajectory at every instant.

Common mistakes

• confusing equations of motion with the equation of the trajectory;
• treating distance traveled as the magnitude of displacement for arbitrary curved motion;
• eliminating time and losing information about motion direction or the valid time interval;
• using a path coordinate without specifying the trajectory, origin, and positive direction.