When a raindrop falls, it increases in size and so its mass...

When a raindrop falls, it increases in size and so its mass at time $t$ is a function of $t$, namely, $m(t) .$ The rate of growth of the mass is $k m(t)$ for some positive constant $k$. When we apply Newton's Law of Motion to the raindrop, we get $(m v)^{\prime}=g m$, where $v$ is the velocity of the raindrop (directed downward) and $g$ is the acceleration due to gravity. The terminal velocity of the raindrop is $\lim _{t \rightarrow x} v(t) .$ Find an expression for the terminal velocity in terms of $g$ and $k$.

Key Concepts

Variable Mass Systems

These are dynamic systems in which the mass changes over time. When analyzing such systems, it is essential to account for the fact that mass is not constant and, therefore, the standard form of Newton's second law must be modified to include the time derivative of the product of mass and velocity. This introduces additional terms in the dynamical equations as the rate of change of mass interacts with other physical quantities.

Newton’s Second Law for Variable Mass

In scenarios where mass varies with time, Newton's second law is applied to the momentum of the system rather than just mass times acceleration. This involves differentiating the momentum (mass times velocity) with respect to time, which requires using the product rule. The resulting equation includes contributions both from the acceleration of the object and from the rate at which its mass changes.

First-order Linear Differential Equations

These equations involve a function and its first derivative, often in the form dy/dt + P(t)y = Q(t). They are commonly solved using integrating factors, which simplify the equation to allow for direct integration. In many physical situations, including decaying or growing processes, such differential equations describe the evolution of dynamic variables like velocity or mass over time.

Terminal Velocity

Terminal velocity is the steady-state velocity an object attains when the net acceleration becomes zero, meaning all forces acting on the object balance each other. In dynamic systems, especially those influenced by forces such as drag or, in variable mass systems, additional momentum effects from mass exchange, the terminal velocity is found by setting the time derivative of velocity to zero and solving for velocity in terms of the given parameters.

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