Particle Velocity

Particle velocity describes the rate of change of particle position and its instantaneous direction of motion. Average velocity describes a finite change of position over a time interval, while instantaneous velocity is obtained by a limiting process and equals the time derivative of the position vector.

Velocity vector

For $\vec r=\vec r(t)$, instantaneous velocity is $\vec v=d\vec r/dt$. The vector $\vec v$ is tangent to the trajectory and points in the direction of particle motion.

Cartesian-coordinate description

For particle motion described by $x(t)$, $y(t)$, and $z(t)$, the velocity components are the time derivatives of the coordinates:

$$v_x=\dot x,\qquad v_y=\dot y,\qquad v_z=\dot z.$$

The speed is:

$$v=\sqrt{v_x^2+v_y^2+v_z^2}.$$

The velocity vector is tangent to the trajectory and points in the direction of motion.

The signs of $v_x$, $v_y$, and $v_z$ indicate how the corresponding coordinates are changing. A zero value of one component does not imply that the particle is at rest.

Path-coordinate description

If position is specified by a path coordinate $s=s(t)$, the algebraic velocity along the trajectory is $v_s=ds/dt$. In vector form, $\vec v=(ds/dt)\vec\tau$, where $\vec\tau$ is the unit tangent vector in the positive $s$ direction.

Average and instantaneous velocity

The average vector velocity over $\Delta t$ is $\Delta\vec r/\Delta t$. It depends on displacement rather than the length of the path traveled. As $\Delta t\to0$, average velocity approaches instantaneous velocity.

Example

For $x=3t^2$ and $y=4t$, the components are $v_x=6t$ and $v_y=4$. At $t=1$ s, the speed is $v=\sqrt{6^2+4^2}=\sqrt{52}\approx7.21$ m/s. The velocity direction coincides with the tangent to the trajectory at that point.

Stopping and reversal

A particle is instantaneously at rest only when its entire velocity vector is zero. In rectilinear motion, a change in the sign of algebraic velocity indicates reversal of direction; an instant with $v=0$ should be interpreted together with the equation of motion.

Common mistakes

• confusing average vector velocity with distance traveled divided by time;
• calculating speed by adding the magnitudes of velocity components;
• ignoring component signs when determining direction;
• assuming that $v_x=0$ means the particle is completely at rest.