Is the acceleration of an object at rest zero?

Our basic question is: if an object is at rest, is its acceleration necessarily zero?

For example, if a car sits at rest its velocity is, by definition, equal to zero. But what about its acceleration?

To answer this question, we will need to look at what velocity and acceleration really mean in terms of the motion of an object. We will use both conceptual and mathematical analyses to determine the correct answer: the object's acceleration is not necessarily zero just because its velocity is zero. This may seem strange at first, but if we unpack it a bit, it should start to make sense.

If we think about the problem quickly, it might seem the acceleration must be zero. At one moment, we're not moving, and a small time later we're still not moving, so there has not been a change in speed. Therefore, the acceleration has to be zero.

There is a mistake here that we can see without doing any calculations.

It's clear that on a regular basis, objects that start out at rest end up in motion. For example, a person standing up from a chair or a plane taking off from a runway. In these cases, there is a clear change from zero velocity to non-zero velocity even though the object starts out at rest. This implies that there must be a moment where the object's acceleration is non-zero although the object remains in the same position.

That was a logical argument for why acceleration in a state of rest must be possible. We can do a better job with a rigorous quantitative argument.

Finding the velocity

Let's start by looking at the object's initial velocity, and confirm that it must be zero. When an object starts from rest (at $x(0)=0$) and starts to accelerate at the rate $\gamma$, its position a time $\Delta T$ later is $x(\Delta T) = \frac12\gamma\left(\Delta T\right)^2.$

With this in hand, we can do a straightforward calculation of the velocity at time zero, $v(0) = \lim_{\Delta T \rightarrow 0} \Delta x/\Delta T$:

$$\begin{aligned} v(0) &= \lim_{\Delta T\rightarrow 0} \frac{\frac12\gamma\left(\Delta T\right)^2 - 0}{\Delta T} \\ &= \lim_{\Delta T\rightarrow 0} \frac12\frac{\gamma\left(\Delta T\right)^2}{\Delta T} \\ &= \lim_{\Delta T\rightarrow 0} \frac12 \gamma \Delta T \\ &= 0 \end{aligned}$$ So that the initial velocity is zero, like we supposed.

Finding the acceleration

Now, let's look at our object's acceleration over the time from the beginning of its acceleration, and a time $\Delta T$ later. At time $t = \Delta T,$ its velocity has increased to $\gamma \Delta T$, thus we have $$\begin{aligned} a(0) &= \lim_{\Delta T\rightarrow 0} \frac{v(\Delta T)- v(0)}{\Delta T} \\ &= \lim_{\Delta T\rightarrow 0} \frac{\gamma \Delta T - 0}{\Delta T} \\ &= \gamma, \end{aligned}$$ which is exactly what we expected to find.

Thus, even though the velocity of an object at rest must be zero, acceleration can clearly be non-zero for objects at rest.