Step 1: Understanding the Concept:
An inertial frame of reference is a coordinate system in which an object experiences zero net force and moves at a constant velocity (or remains at rest), meaning Newton's first law of motion is fully valid.
Step 2: Key Formula or Approach:
The approach involves Galilean transformations. If the position in frame $S$ is $\vec{r}$ and in frame $S'$ is $\vec{r}' = \vec{r} - \vec{v}t$ (where $\vec{v}$ is constant relative velocity), taking the second derivative with respect to time gives the acceleration relationship.
Step 3: Detailed Explanation:
Let frame $S$ be an established inertial frame.
Suppose frame $S'$ is moving with a constant velocity $\vec{v}$ relative to frame $S$.
Because the relative velocity is constant, the derivative of velocity with respect to time is zero. Hence, the relative acceleration between the frames is zero.
If an object has an acceleration $\vec{a}$ measured in frame $S$, its acceleration $\vec{a}'$ measured in frame $S'$ will be exactly the same ($\vec{a}' = \vec{a}$).
If no forces act on an object, $\vec{a} = 0$, which means $\vec{a}' = 0$ as well.
Since Newton's first law holds true in frame $S'$ just as it does in frame $S$, frame $S'$ is also classified as an inertial frame.
Step 4: Final Answer:
The correct option is (C).