A

Step 1: Identify the system. The gas is confined within an insulated container and at rest with respect to the block. Therefore, its center of mass does not move.
Step 2: Evaluate momentum with respect to block. Since the gas is at rest in the insulated container, the average linear momentum with respect to the block is zero.
Step 3: Evaluate average linear momentum with respect to Earth. As the gas is not moving, its total momentum with respect to Earth is also zero.
Step 4: Calculate rms speed of gas. The rms speed of gas with respect to Earth is zero because the gas does not move in the Earth frame.
Step 5: Therefore, the average linear momentum and rms speed can be defined and matched:
- Average linear momentum w.r.t. block: [A] Zero
- Average linear momentum w.r.t. Earth: [B] Zero
- v_rms w.r.t. block: [C] \sqrt{\frac{3kT}{m}}
- v_rms w.r.t. Earth: [D] Zero
Matches are:
(i) Average linear momentum w.r.t. block: [A] Zero
(ii) v_rms of the gas w.r.t. earth: [B] Zero
(iii) Average linear momentum w.r.t. Earth: [C] Zero
(iv) v_rms w.r.t. block: [D] \sqrt{\frac{3kT}{m}}
Hence, the options match correctly.