Acceleration

Acceleration tells us how an object's velocity changes over time. It answers two important questions: How fast is an object's speed changing? And in which direction?

Visualizing Motion

Let's see how acceleration works with a moving object:

In this example, the object starts with $v_0 = 1$ meter per second. It begins moving right and after $3$ seconds it attains a velocity of $v_f=7$ meters per second. The acceleration of the object is given by:

$\vec{a} = \dfrac{\Delta \vec{v}}{\Delta t} = \dfrac{7\text{ m/s}-1\text{ m/s}}{3\text{s} - 0\text{ s}} = \dfrac{6 \text{m/s} }{3\text{s}} = 2 \text{m/s}^2$ in the positive direction.

Note: In 1D, like for displacement, we may sometimes drop the "direction", as well as the arrows above $a$ and $\Delta v$, since the direction should be clear from whether it is a positive or negative value, so long as we've labelled our coordinate system appropriately.

Direction Matters

Acceleration is a vector pointing in the direction of $\Delta v$, not necessarily in the direction of motion. The sign tells us in which direction the object's velocity is changing:

• Positive Acceleration $(a>0)$: Velocity gets more positive.
• Negative Acceleration $(a<0)$: Velocity gets more negative.

An object can have negative acceleration while moving forward (slowing down) or negative acceleration while moving backward (speeding up in the negative direction).

For example, if an object's velocity changes from $v_i=-9$ m/s to $v_f=-3$ m/s,