How to Solve the Differential Equation dv/dt g - (c/m)v: A Comprehensive Guide

Understanding the solution of the differential equation (frac{dv}{dt} g - frac{c}{m} v) is crucial in various fields such as physics and engineering. This guide will walk you through the process using both the method of separation of variables and recognizing it as a first-order linear ordinary differential equation. We will also explore the concept of terminal velocity.

Solving the Differential Equation Using Separation of Variables

To solve the given differential equation, we start with the rearrangement of the equation:

(frac{dv}{dt} - frac{c}{m} v g)

We can separate the variables to get:

(frac{dv}{g - frac{c}{m} v} dt)

Integrating both sides, we get:

(int frac{dv}{g - frac{c}{m} v} int dt)

The left-hand side can be integrated as:

(int frac{dv}{g - frac{c}{m} v} -frac{m}{c} ln |g - frac{c}{m} v|)

This simplifies to:

(-frac{m}{c} ln |g - frac{c}{m} v| t C)

where C is the constant of integration. Solving for v, we get:

(ln |g - frac{c}{m} v| -frac{c}{m} t C')

Exponentiating both sides, we obtain:

(|g - frac{c}{m} v| e^{-frac{c}{m} t C'} e^{C'} e^{-frac{c}{m} t})

Letting (C' ln A) for some constant A, we get:

(g - frac{c}{m} v A e^{-frac{c}{m} t})

Solving for v, we have the general solution:

(v frac{m}{c} g - frac{m}{c} A e^{-frac{c}{m} t})

This is the final general solution, where A is a constant determined by initial conditions.

Solving as a First-Order Linear Ordinary Differential Equation

We can also recognize the differential equation as a first-order linear ordinary differential equation of the form:

(frac{dv}{dt} frac{c}{m} v g)

The method of integrating factor can be used here. The integrating factor is given by:

(mu(t) e^{int frac{c}{m} dt} e^{frac{c}{m} t})

Multiplying both sides of the differential equation by the integrating factor, we get:

(e^{frac{c}{m} t} frac{dv}{dt} frac{c}{m} e^{frac{c}{m} t} v g e^{frac{c}{m} t})

The left-hand side can be rewritten as the derivative of a product:

(frac{d}{dt} left(v e^{frac{c}{m} t} right) g e^{frac{c}{m} t})

Integrating both sides, we get:

(v e^{frac{c}{m} t} int g e^{frac{c}{m} t} dt)

The right-hand side can be integrated as:

(int g e^{frac{c}{m} t} dt g cdot frac{m}{c} e^{frac{c}{m} t} C)

Solving for v, we have:

(v e^{frac{c}{m} t} frac{g m}{c} e^{frac{c}{m} t} C)

Dividing both sides by (e^{frac{c}{m} t}), we get the general solution:

(v(t) frac{g m}{c} C e^{-frac{c}{m} t})

Concept of Terminal Velocity

The concept of terminal velocity is important in the context of this differential equation. As time (t) approaches infinity, the term (C e^{-frac{c}{m} t}) approaches zero, and the velocity (v(t)) converges to:

(lim_{t to infty} v(t) frac{g m}{c})

This velocity is the terminal velocity, which is the constant velocity that an object achieves when the force of gravity is balanced by the drag force.

Conclusion

Understanding how to solve the differential equation (frac{dv}{dt} g - frac{c}{m} v) is crucial for modeling various physical phenomena, such as the motion of falling objects or charged particles. Whether through the method of separation of variables or by recognizing it as a first-order linear differential equation, the solution provides valuable insights into the behavior of such systems.

Keywords

differential equation, terminal velocity, first-order linear differential equation