Question

A raindrop falls under gravity and captures water molecules from the atmosphere. Its mass changes at the rate $\lambda m(t)$, where $\lambda$ is a positive constant and $m(t)$ is the instantaneous mass. Assume gravity is constant and captured water is at rest before capture. Which of the following statements is correct?

• A. The speed of the raindrop increases linearly with time.
• B. The speed of the raindrop increases exponentially with time.
• C. The speed of the raindrop approaches a constant value when $\lambda t \gg 1$.
• D. The speed of the raindrop approaches a constant value when $\lambda t \ll 1$.

Hint:

Whenever mass increases proportionally to itself, the velocity–time behaviour usually involves an exponential decay term.

Answer 1

The Correct Option is C

Step 1: Use momentum conservation for variable mass systems.
The equation of motion when mass increases by capturing matter at rest is $\displaystyle m\frac{dv}{dt} = mg - v\frac{dm}{dt}.$ Given $\displaystyle \frac{dm}{dt} = \lambda m$, substitute to obtain $m\frac{dv}{dt} = mg - \lambda m v.$

Step 2: Simplify the differential equation.
Cancelling $m$, $\displaystyle \frac{dv}{dt} = g - \lambda v.$

Step 3: Solve the DE.
Solution of $\displaystyle \frac{dv}{dt}+\lambda v=g$ is $\displaystyle v(t)=\frac{g}{\lambda}\left(1-e^{-\lambda t}\right).$

Step 4: Analyze behaviour.
As $t\to\infty$ (i.e. $\lambda t \gg 1$), $e^{-\lambda t}\to 0$ and $v\to \dfrac{g}{\lambda}$ (a constant terminal speed).

Step 5: Conclusion.
The raindrop reaches a constant speed at large times $\lambda t\gg 1$, matching option (C).