COMEDK · Physics · 3. Motion In One Dimension

The displacement \(x\) of a particle varies with time \(t, x=a e^{-p t}+b e^{q t}\), where \(a, b, p\) and \(q\) are positive constant. The velocity of the particle will

A go on increasing forever
B be independent of \(p\) and \(q\)
C drop to zero when \(p=q\)
D go on decreasing with time

Verified Solution

Answer & Solution

Correct Answer

(A) go on increasing forever

Step-by-step Solution

Detailed explanation

The displacement of the particle is given by \(x = a e^{-pt} + b e^{qt}\), where \(a, b, p, q > 0\). The velocity \(v\) is the time derivative of displacement \(x\): \(v = \dfrac{dx}{dt} = \dfrac{d}{dt}(a e^{-pt} + b e^{qt})\) \(v = -ap e^{-pt} + bq e^{qt}\) To determine the…

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.

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