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Step-by-Step Solution
Step 1: Identify the Physical Situation
A lift is moving downward with acceleration $a$. A man inside the lift drops a ball. We want to determine the acceleration of the ball as observed by:
The man inside the lift.
Another observer standing on the ground (an inertial frame).
Step 2: Acceleration of the Ball (Inertial Frame)
From the perspective of a stationary observer on the ground, there is no additional upward or downward acceleration other than gravity acting on the ball (neglecting air resistance). Thus, in the ground frame, the ball simply accelerates downward with acceleration:
$$g.$$
Step 3: Acceleration of the Ball (Non-inertial Frame)
Inside the lift, which is accelerating downward with acceleration $a$, the reference frame of the man is non-inertial. In a non-inertial reference frame, we account for a “pseudo-force” acting on objects if we wish to use Newton’s second law in that frame. The effective (or relative) acceleration of the ball with respect to the man in the lift is computed as follows:
Let
$$\vec{a}_b$$
be the ball’s actual acceleration (with respect to the ground), and
$$\vec{a}_m$$
be the acceleration of the man (and the lift) with respect to the ground.
The relative acceleration of the ball with respect to the man is:
$$
\vec{a}_{\text{ball,man}} = \vec{a}_b - \vec{a}_m.
$$
Since
$$\vec{a}_b = g$$
(downward) and
$$\vec{a}_m = a$$
(downward), their difference (taking downward as positive) is:
$$
a_{\text{ball,man}} = g - a.
$$
Step 4: Conclude the Observed Accelerations
Hence, the accelerations of the ball as observed by:
The man in the lift:
$$g - a.$$
The observer on the ground:
$$g.$$
This matches the correct answer: $(g - a, \, g)$.
