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 $t, m(t) .$ 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 $\lim _{t \rightarrow \infty} v(t) .$ Find an expression for the terminal velocity in terms of $g$ and $k$

Key Concepts

Newton's Second Law for Variable Mass Systems

Newton's second law states that the rate of change of momentum equals the net force acting on a body. For systems with changing mass, the momentum derivative must account for both the change in velocity and the change in mass. This requires applying the product rule for differentiation to the momentum, which is essential when analyzing bodies like raindrops that gain mass as they fall.

Exponential Growth

Exponential growth occurs when the rate of change of a quantity is directly proportional to the current amount of that quantity. In the context of a falling raindrop, this concept describes how the mass increases over time at a rate proportional to its own value, leading to an exponential growth in mass.

First Order Linear Differential Equations

First order linear differential equations have the form dy/dt + P(t)y = Q(t) and are typically solved using integrating factors. In this problem, the differential equation that describes the velocity, derived from Newton's law with variable mass, is a first order linear ODE whose steady state solution provides the terminal velocity.

Terminal Velocity

Terminal velocity is the constant speed that an object eventually reaches when the force of gravity is balanced by other forces or effects, leading to zero acceleration. For a falling raindrop with increasing mass, terminal velocity is determined by finding the long-term behavior of the velocity as described by the differential equation.

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