A particle is moving along the \(x-\)axis with its coordinate with the time '\(t\)' given be \(\mathrm{x}(\mathrm{t})=10+8 \mathrm{t}-3 \mathrm{t}^{2} .\) Another particle is moving the \(y-\)axis with its coordinate as a function of time given by \(\mathrm{y}(\mathrm{t})=5-8 \mathrm{t}^{3} .\) At \(\mathrm{t}=1\; \mathrm{s},\) the speed of the second particle as measured in the frame of the first particle is given as \(\sqrt{\mathrm{v}} .\) Then \(\mathrm{v}\) (in \(\mathrm{m} / \mathrm{s})\) is

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.