Defining Directions for Position, Velocity, and Acceleration

I teach an engineering dynamics course this term. I have read several engineering dynamics textbooks and am not impressed by how thick these books are. They gave a lot of examples and problems, but clear concise exposition of concept is often lacking. Reading these books makes me want to write my own textbook.

One example is a lack of clear explanation on how to consistently assign and interpret sign values (positive or negative) to position, velocity, and acceleration. An easy way to get rid of this confusion is offered here. I hope it is concise enough.

Velocity is the rate of change of position x of a particle with time t:

v = dx/dt.

The direction (i.e., sign) definition of the three quantities starts with the definition for the position x. If dx > 0, then v > 0 since dt is always positive. Now, dx = x(final) – x(initial) > 0.

v > 0 thus produces x(final) > x(initial). In other words, v > 0 corresponds to the direction of increasing x.

What about the acceleration? The acceleration a depends on velocity v through this relation:

a = dv/dt,

so that a > 0 if dv > 0 since dt is always positive. By the same argument, we say that a > 0 produces v(final) > v(initial).

It is possible that both v(final) and v(initial) < 0, even though v(final) > v(initial), so that a > 0 points in one direction, while both final and initial velocities point in the opposite direction. Having a > 0 or a < 0 thus does not correspond to a direction of the particle's motion. The direction of the motion is determined by v.

If a has an opposite sign to v, then the particle's speed will slow down. If a and v share the same direction, then the particle's speed increases.