The mass m of a raindrop increases as it picks up moisture...

The mass $m$ of a raindrop increases as it picks up moisture during its vertical descent through still air. If the air resistance to motion of the drop is $R$ and its downward velocity is $v,$ write the equation of motion for the drop and show that the relation $\Sigma F=d(m v) / d t$ is obeyed as a special case of the variable-mass equation.

Key Concepts

External Forces and Air Resistance

In dynamics, all forces acting on the system must be accounted for. Air resistance, often modeled as a force proportional to velocity (and opposite in direction), is one such external force. In variable-mass problems, these external forces, including drag or air resistance, are summed to equal the rate of change of momentum. This ensures that all factors influencing the system's motion—both dynamical (velocity) and configurational (mass change)—are integrated into the equation of motion.

Momentum and Its Differential

The rate of change of momentum, given by d(mv)/dt, captures the complete dynamics in systems where both mass and velocity vary with time. The derivative expands to m·(dv/dt) + v·(dm/dt), with the second term explicitly representing the contribution from mass changing over time. This formulation is key to understanding how external forces result in both acceleration and mass variation effects.

Variable Mass Systems

This concept deals with systems in which the mass of the object is not constant over time. In such problems, the standard form of Newton’s second law (F = ma) must be modified to account for the change in mass. The momentum of the system is given by m·v, and its time derivative includes not only the acceleration term m·(dv/dt) but also a term due to the rate of change of mass (dm/dt) times the velocity. This modification is essential when dealing with situations like a raindrop growing by collecting moisture.

Newton's Second Law for Variable Mass

Newton’s second law in its traditional form assumes constant mass; however, for variable-mass systems the force must be equated to the time rate of change of momentum, expressed as d(mv)/dt. This form inherently includes contributions from both changes in velocity and changes in mass, ensuring that the dynamics are accurately captured even when mass is accreting or being depleted.

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