Differential Equations: Problem Solving

When analyzing the motion of falling objects, it is essential to consider not only the force of gravity but also the opposing force of air resistance. A practical example involves releasing a heavy test weight during a safety check on a ship. As the weight falls from rest, gravity accelerates it downward while air resistance exerts an upward force that increases with velocity. This dynamic interplay of forces is well described by differential equations, which provide a mathematical framework for modeling the object’s changing velocity over time.

Forces and Differential Modeling

According to Newton’s Second Law, the net force on the falling weight determines its acceleration. Gravity exerts a constant force equal to the object's mass times the gravitational acceleration, while air resistance is typically modeled as being proportional to the object’s velocity. When these forces are combined, the net force yields a first-order differential equation relating the rate of change of velocity to the velocity itself.

Exponential Behavior and Terminal Velocity

Solving the resulting differential equation gives a velocity function that increases over time but asymptotically approaches a finite limit. This behavior reflects the gradual balancing of gravity and air resistance, culminating in a state known as terminal velocity—the point at which acceleration ceases, and the object falls at a constant speed. With a mass of 10 kg and a drag constant of 2 N·s/m, the calculated terminal velocity is 49 m/s. This outcome illustrates how differential equations effectively model real-world motion and reveal the role of air resistance in limiting acceleration during free fall.