Problem 12-15: Engineering Mechanics- Dynamics (Hibbeler 14th Edition)

Rectilinear Kinematics with Velocity-Dependent Deceleration

This lesson derives velocity and position as explicit functions of time for a particle subjected to a deceleration proportional to the cube of its velocity, $a = -kv^3$. The method uses separation of variables and direct integration of the kinematic differential equations, with verification of initial conditions and the original acceleration law.

Question:
A particle is moving with a velocity of v-naught when s equals zero and t equals zero. If it is subjected to a deceleration of a equals negative k times v cubed, where k is a constant, determine its velocity and position as functions of time.

In this lesson, you will learn to:
• Set up and separate the kinematic differential equation $a = dv/dt$ when acceleration depends on velocity.
• Apply definite integration with initial conditions to obtain $v(t)$ in closed form.
• Use