Step 1: Understanding the Concept:
An inertial frame of reference is defined as a frame of reference that is not undergoing any acceleration.
In such a frame, Newton's first law of motion (the law of inertia) holds true without the need for fictitious forces.
A non-inertial frame is one that is accelerating or rotating.
Step 2: Key Formula or Approach:
The defining condition for an inertial frame is that the acceleration vector is zero:
\[ \vec{a} = \frac{d\vec{v}}{dt} = 0 \]
This means the velocity vector \(\vec{v}\) must be constant in both magnitude (speed) and direction.
Step 3: Detailed Explanation:
Let's systematically evaluate each given option based on the condition \(\vec{a} = 0\):
(A) A train slowing down to stop is experiencing deceleration. Because its speed is changing, \(\vec{a} \neq 0\). This is a non-inertial frame.
(B) A merry-go-round revolves in a circular path. Even if the speed is constant, the direction of motion is continuously changing, resulting in a centripetal acceleration. Because \(\vec{a} \neq 0\), this is a non-inertial frame.
(C) A bus moving with a constant velocity has both constant speed and a constant direction of motion in a straight line. Here, \(\vec{a} = 0\). This perfectly describes an inertial frame.
(D) An aeroplane taking off is increasing its speed to gain lift. Because its speed is changing rapidly, \(\vec{a} \neq 0\). This is a non-inertial frame.
Step 4: Final Answer:
The driver in a bus moving with constant velocity is in an inertial frame of reference.