2.1.1 Sign Convention

Physicists use three quantities to describe motion: displacement, velocity and acceleration.

Displacement s denotes a position by its distance and direction from some reference point.

Velocity v is the rate of change of displacement.

\displaystyle v=\frac{{ds}}{{dt}}

Acceleration a is the rate of change of velocity.

\displaystyle a=\frac{{dv}}{{dt}}

In general s v and a are all vector quantities, and have to be analyzed as vectors (meaning vector diagrams, vector algebra, etc). Luckily, for rectilinear motion, s v and a can be “scalarized” (meaning we just add or subtract them as if they are scalar quantities) if we adopt a sign convention.

Let me elaborate. Rectilinear motion refers to motion which stays along a straight line. Such as motion has only two possible directions (e.g. forward vs backward, upward vs downward, rightward vs leftward etc). So, we can use the positive sign to denote one direction, and the negative sign for the other. For example, if we use the positive sign to denote the northward direction, and the negative sign for the southward direction, we have adopted a “north is positive” sign convention.

For example, +2 m and −2 m may denote displacements at 2 m east and west (of the origin) respectively. You may want to think of 2 m as the distance, and the +- signs as the direction.

Also, +2 m s-1 and −2 m s-1 may denote 2 m s-1 in the forward and backward directions respectively. You may want to think of 2 m s-1 as the speed, and the +- signs as the direction.

Similarly, +2 m s-2 and −2 m s-2 may denote acceleration in the rightward and leftward directions respectively.

Did you notice that a negative acceleration does not necessarily imply deceleration? Because the negative sign merely means that the velocity (not speed) is changing in the leftward direction. If you are currently moving rightward, then a leftward acceleration means you are slowing down. But if you are currently moving leftward (or at rest), then a leftward acceleration means you are speeding up.

Instead, the motion speeds up if the acceleration has the same sign as the velocity (so a and v are in the same direction), and slows down if the acceleration has the opposite sign as the velocity (so a and v are in opposite directions).

Video Explanation

Sign Convention

Concept Test

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