91Ó°ÊÓ

Understanding the rate of change is essential when dealing with differential equations. Think of the rate of change as how fast something is moving or varying over time. In this context, we're examining the velocity of an object and how it changes. The differential equation given is: \( \frac{dv}{dt} = -g - \frac{c}{m}v \). Here, \( \frac{dv}{dt} \) represents the rate at which the velocity \( v \) changes over time. Understanding this rate is crucial as it helps us determine how quickly the object's velocity is decreasing due to gravity (indicated by -g) and a resistive force proportional to velocity (indicated by \( -\frac{c}{m}v \)).

The negative signs indicate that both gravity and resistance are acting in a direction opposite to the object's motion. Summarizing it:

• -g: the constant rate due to gravity.
• -\frac{c}{m}v: the rate of change due to resistive force (like friction or air resistance).

The end goal is to understand how these factors balance out to find the equilibrium condition.

The equilibrium condition is reached when the rate of change of velocity becomes zero. In simpler terms, this means the object has reached a state where its velocity does not change anymore. For any differential equation involving time, an equilibrium condition helps determine a stable solution.

For the given differential equation \( \frac{dv}{dt} = -g - \frac{c}{m}v \), the equilibrium condition can be found by setting the rate of change \( \frac{dv}{dt} \) to zero. This is because, at equilibrium, the velocity is no longer changing:

• \( 0 = -g - \frac{c}{m}v \).

Solving for the equilibrium velocity \( v \), you'll get:

\ \frac{c}{m}v = -g \
\ v = -\frac{mg}{c} \

This value of \( v \) tells us the steady-state (or equilibrium) velocity of the object when the forces of gravity and resistance have balanced out.

The limiting velocity is essentially the equilibrium velocity in the context of motion through a resistive medium. It represents the maximum velocity an object can achieve when falling under the influence of gravity and resistance. Another term often used interchangeably is 'terminal velocity'.

From the steps in the exercise, we determined that the limiting velocity \[v_{lim} = -\frac{mg}{c} \]. This result indicates:

• When an object is falling, the downward force of gravity \ (mg) \ is balanced by the upward resistive force \( \frac{cv}{m} \).
• At this point, the object stops accelerating and continues to fall at a constant velocity.

Understanding this helps in various physical scenarios, such as predicting how fast a parachutist might fall before deploying their parachute.

So to wrap it up, the key takeaway is that the limiting velocity is the steady speed an object reaches due to a balance of all acting forces, illustrating the concept of equilibrium in motion.