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The displacement x of a particle varies with time t as x = ae⁻αt + be⁻βt, where a, b, α and β are positive constants. The velocity of the particle will:

1. be independent of α and β
2. drop to zero when α = β
3. go on decreasing with time
4. go on increasing with time

Correct answer: go on decreasing with time

Solution

The velocity is the time derivative of displacement. Differentiating x = ae⁻αt + be⁻βt gives v = -aαe⁻αt - bβe⁻βt. Since both terms decrease exponentially with time due to the negative exponents, the velocity will also decrease with time.

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