Particle Acceleration

Particle acceleration describes the rate of change of the velocity vector. Acceleration can arise from a change in speed, a change in velocity direction, or both.

Acceleration vector

Instantaneous acceleration is $\vec a=d\vec v/dt=d^2\vec r/dt^2$. Unlike velocity, the acceleration vector is not generally tangent to the trajectory.

Cartesian and path components

In Cartesian coordinates, acceleration components are the second time derivatives of the coordinates:

$$a_x=\ddot x,\qquad a_y=\ddot y,\qquad a_z=\ddot z.$$

Using path coordinates:

$$a_\tau=\frac{dv}{dt},\qquad a_n=\frac{v^2}{\rho},$$

where $\rho$ is the radius of curvature of the trajectory. The normal acceleration points toward the center of curvature.

Tangential acceleration

The component $a_\tau=dv/dt$ describes the change in speed. When the tangential acceleration points with the velocity, speed increases; when it points opposite the velocity, speed decreases.

Normal acceleration

The component $a_n=v^2/\rho$ results from a change in velocity direction. It always points toward the center of curvature and vanishes for rectilinear motion, for which the radius of curvature is formally infinite.

Total acceleration

Because tangential and normal components are perpendicular, the acceleration magnitude is $a=\sqrt{a_\tau^2+a_n^2}$. Its direction follows from $\vec a=a_\tau\vec\tau+a_n\vec n$.

Example

A particle moves on a circle of radius $\rho=2$ m at a speed of 6 m/s that is increasing at 3 m/s². Then $a_\tau=3$ m/s², $a_n=6^2/2=18$ m/s², and $a=\sqrt{3^2+18^2}\approx18.25$ m/s².

Common mistakes

• assuming acceleration is zero whenever speed is constant;
• directing normal acceleration along the tangent;
• confusing the radius of curvature with distance to an arbitrary coordinate origin;
• adding $a_\tau$ and $a_n$ algebraically when calculating total acceleration magnitude.