{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation 


m · v'(t) = mg + f(v)
mass | acceleration | external forces


where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.


c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a. 

Verified step by step guidance

1. Start with the given differential equation: \(m \cdot v'(t) = mg + f(v)\), where \(f(v) = -kv^2\). Substitute \(f(v)\) to get \(m \cdot v'(t) = mg - kv^2\).
2. Rewrite the equation to isolate \(v'(t)\): \(v'(t) = g - \frac{k}{m} v^2\). This is a separable differential equation.
3. Separate variables by writing \(\frac{dv}{dt} = g - \frac{k}{m} v^2\) as \(\frac{dv}{g - \frac{k}{m} v^2} = dt\).
4. Integrate both sides: integrate the left side with respect to \(v\) and the right side with respect to \(t\). The integral on the left involves a rational function of \(v^2\), which can be handled using a trigonometric substitution or recognizing it as a standard integral form.
5. Apply the initial condition \(v(0) = 0\) to solve for the constant of integration after performing the integration, and express \(v\) explicitly as a function of \(t\).

Key Concepts

Newton's Second Law and Differential Equations

Newton's Second Law states that mass times acceleration equals the sum of external forces. In this problem, it leads to a differential equation relating velocity and time. Understanding how to translate physical laws into differential equations is essential for modeling motion under forces like gravity and air resistance.

Separable Differential Equations

A separable differential equation can be written so that all terms involving the dependent variable are on one side and all terms involving the independent variable are on the other. This allows integration on both sides to find the solution. Recognizing and solving separable equations is key to finding velocity as a function of time here.

Modeling Air Resistance as a Quadratic Drag Force

Air resistance is often modeled as a force proportional to the square of velocity, expressed as f(v) = -kv², where k > 0 is the drag coefficient. This nonlinear term affects the acceleration and velocity, making the differential equation nonlinear. Understanding this model helps in setting up and solving the equation for velocity under drag.