Problem 12-27: Engineering Mechanics- Dynamics (Hibbeler 14th Edition)

Solving a Separable ODE for Terminal Velocity Kinematics

This lesson applies separation of variables and partial fractions to solve a first-order ODE modeling air-resistance drag, where acceleration decreases from g to zero as a particle approaches terminal velocity. Students find the explicit time required to reach a given fraction of terminal velocity starting from rest.

Question:
A particle falls through air with acceleration given by a equals g over v-f squared, times the quantity v-f squared minus v squared, where g is gravitational acceleration and v-f is the terminal velocity. The particle starts from rest. Determine the time t needed for the velocity to reach v equals v-f divided by 2.

In this lesson, you will learn to:
• Set up and classify a separable first-order ODE from a physical acceleration model
• Apply partial fraction decomposition to integrate a rational function of