What Does a Negative Sign on Acceleration Mean?

A negative sign on acceleration tells you its direction relative to the coordinate system you chose. It does not, by itself, tell you whether an object is slowing down. To determine that, compare the signs of velocity and acceleration.

What the sign means

In one-dimensional motion, acceleration is the rate at which velocity changes:

$$a=\frac{dv}{dt}.$$

If you choose the positive direction to be right, then a negative acceleration points left. If you instead define left as positive, the same physical acceleration has a positive component. The sign depends on your axis choice; the direction of the physical acceleration does not.

Acceleration is a vector. In one dimension, its signed value records direction, while its magnitude is nonnegative: for example, an acceleration of $-3\ \text{m/s}^2$ has magnitude $3\ \text{m/s}^2$ and points along the negative axis.

Use velocity to tell whether speed changes

Speed is the magnitude of velocity. In one dimension, the relationship is:

• Velocity and acceleration have opposite signs: speed decreases.
• Velocity and acceleration have the same sign: speed increases.

This follows because acceleration changes velocity. If the velocity is positive and acceleration is negative, the positive velocity component moves toward zero. If both are negative, the velocity becomes more negative and its magnitude grows.

At the instant an object’s velocity is zero, this sign test needs care: the object may be turning around, or it may simply be at rest. Look at the motion just before and after that instant.

Worked example: a ball thrown upward

Choose upward as positive. A ball moving upward has positive velocity, but gravity points downward, so its acceleration is negative:

$$v=+12\ \text{m/s},\qquad a=-9.8\ \text{m/s}^2.$$

The signs are opposite, so the ball’s upward speed decreases. Ignoring air resistance, after one second its velocity is approximately

$$v=12+(-9.8)(1)=2.2\ \text{m/s}.$$

At the top of its path, its velocity is momentarily zero, but its acceleration is still $-9.8\ \text{m/s}^2$. The ball then moves downward. Its velocity becomes negative, matching the acceleration’s sign, so its downward speed increases. The acceleration has not changed sign at the top; the velocity has.

Common mistakes

Assuming negative acceleration means slowing down

An object moving in the negative direction with negative acceleration speeds up. For example, if a car has $v=-5\ \text{m/s}$ and $a=-2\ \text{m/s}^2$, its velocity becomes more negative, so its speed increases.

Treating “deceleration” as a synonym for negative acceleration

Deceleration usually means that speed is decreasing. That can happen with positive or negative acceleration, depending on the direction of motion. In one-dimensional motion, the acceleration opposes the velocity when the object is slowing down.

Forgetting to state the positive direction

A sign has no directional meaning until you define the coordinate axis. Write down your convention before substituting values into equations. If you reverse the axis, the signs of the velocity and acceleration components reverse too, but the physical motion is unchanged.

Confusing acceleration with its magnitude

The signed acceleration can be negative, while its magnitude cannot. A magnitude describes how large the acceleration is; the sign of a component describes which way it points along the chosen axis.

A quick sign check

For motion along a straight line, first assign positive and negative directions. Then identify the signs of velocity and acceleration at the time you care about. Opposite signs mean the velocity component is moving toward zero; matching signs mean it is moving away from zero. If velocity crosses zero, analyze the motion on each side of the crossing separately.