8.6
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 safety check on a ship uses a heavy test weight. The weight is lifted and then released to study how air resistance affects motion. Once it is let go, the weight starts from rest and falls through the air.
Gravity pulls it downward, while the air pushes upward against its motion. According to Newton’s Second Law, the change in speed depends on the net force.
Combining these forces gives a differential equation that links acceleration to speed. Dividing the equation by mass gives a simpler form.
Defining the ratio of the drag constant to the mass as the constant b makes the differential equation easier to separate.
Integrating the equation and rewriting it to find the equation for velocity as a function of time gives an exponential equation. Using the initial velocity of zero helps find the remaining constant in the solution.
As time increases, the velocity approaches a constant value known as terminal velocity. With a weight of 10 kilograms and a drag constant of 2 newton-seconds per meter, the model predicts a terminal velocity of 49 meters per second.
View the full transcript and gain access to JoVE Core videos
Q1: How do gravity and air resistance interact in a falling object?
Gravity pulls the falling object downward with constant force, while air resistance pushes upward and increases with velocity. According to Newton's Second Law, the net force from these opposing forces determines the object's acceleration. This dynamic interplay is described by modeling with differential equations, which relate the rate of change of velocity to the velocity itself.
Q2: What is terminal velocity and when does it occur?
Terminal velocity is the constant speed an object reaches when gravity and air resistance balance completely, causing acceleration to cease. As time increases, the falling object's velocity asymptotically approaches this finite limit. For a 10-kilogram weight with a drag constant of 2 newton-seconds per meter, the terminal velocity is 49 meters per second.
Q3: Why is the drag constant important in modeling falling motion?
The drag constant quantifies how strongly air resistance opposes motion relative to the object's mass. Dividing the drag constant by mass creates a simplified constant that makes the differential equation easier to separate and solve. This parameter directly determines the terminal velocity and how quickly the object approaches it during free fall.
Q4: How does solving a differential equation reveal velocity over time?
Integrating the differential equation that links acceleration to speed yields an exponential velocity function. Using the initial condition of zero velocity helps determine the remaining constant in the solution. This exponential equation shows how velocity increases from rest and gradually approaches terminal velocity as time progresses.
Q5: What role does Newton's Second Law play in setting up the motion equation?
Newton's Second Law states that the net force equals mass times acceleration, establishing the fundamental relationship between forces and motion. By combining gravitational force and air resistance force, this law produces a first-order differential equation relating acceleration to velocity. Dividing by mass simplifies the equation into a more manageable form for solving.
Q6: Why is the initial velocity condition essential when solving this differential equation?
The initial velocity condition of zero specifies the starting state of the falling weight at the moment of release. When substituted into the integrated solution, it determines the unknown constant in the exponential velocity equation. Without this boundary condition, the solution would contain an arbitrary constant and could not predict the specific motion of the weight.
Q7: How does the exponential behavior of velocity reflect real-world falling motion?
The exponential velocity function shows rapid acceleration initially when air resistance is weak, then gradual slowing as resistance increases with speed. This asymptotic approach to terminal velocity mirrors actual falling objects, where acceleration decreases over time. The mathematical model effectively captures how air resistance progressively limits acceleration during free fall.