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The displacement of a particle is given by x = a*e^(-alpha*t) + b*e^(beta*t), where a, b, alpha, and beta are all positive constants. Which of the following correctly describes the velocity of the particle?

1. The velocity goes on decreasing with time
2. The velocity is independent of alpha and beta
3. The velocity drops to zero when alpha = beta
4. The velocity goes on increasing with time

Correct answer: The velocity goes on increasing with time

Solution

v = dx/dt = -a*alpha*e^(-alpha*t) + b*beta*e^(beta*t). At t=0: v = -a*alpha + b*beta (can be positive or negative). As t increases: -a*alpha*e^(-alpha*t) -> 0 (goes toward zero from its negative value), and b*beta*e^(beta*t) -> +inf. So velocity continually increases with time (eventually dominated by the positive growing exponential). Option D is correct. When alpha=beta: v = e^(-alpha*t)*(-a*alpha) + e^(beta*t)*(b*beta) = alpha*(-a*e^(-alpha*t) + b*e^(beta*t)), which is not zero in general. Option C is wrong.

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