Question

An object with mass \(m\) is given an initial velocity \(v_{0} \leq 0\) in a medium that exerts a resistive force of the form \(a|v| /(1+|v|),\) where \(a\) is positive constant. (a) Set up a differential equation for the speed of the object. (b) Use your favorite numerical method to solve the equation you found in (a), to convince yourself that there's a unique number \(a_{0}\) such that \(\lim _{t \rightarrow \infty} s(t)=\infty\) if \(a \leq a_{0}\) and \(\lim _{t \rightarrow \infty} s(t)\) exists (finite) if \(a>a_{0}\). (We say that \(a_{0}\) is the bifurcation value of \(a\).) Try to find \(a_{0}\) and \(\lim _{t \rightarrow \infty} s(t)\) in the case where \(a>a_{0} .\) HINT: See Exercise \(14 .\)

Short answer

Answer: The differential equation that models the motion of the object in the medium is \( m \frac{dv}{dt} = -a \frac{|v|}{1 + |v|} \).

Step-by-step solution

Set up the differential equation

To set up a differential equation for the speed of the object, we need to find the equation of motion. This can be done by applying Newton's second law, which states that the net force acting on an object is equal to the product of its mass and acceleration. The net force on the object is the resistive force, so we have \(F_{net} = F_{resistive}\). The resistive force is given by \(a|v| /(1+|v|)\). Therefore, we can write Newton's second law as: \[ m \frac{dv}{dt} = -a \frac{|v|}{1 + |v|} \]

Separation of Variables

To solve the differential equation, we will use the method of separation of variables. First, move all terms with v on one side and terms with t on the other side: \[ \frac{m}{a} \frac{dv}{1 + |v|} = - |v| dt \] Now integrate both sides: \[\frac{m}{a} \int \frac{dv}{1+|v|} = \int (- |v|) dt \]

Solve the Integral

To solve the integral, we will use substitution. Let \(u = 1+|v|\). Thus, \(du = dv\) for \(v\leq0\). Then, the integral on the left side becomes: \[ \frac{m}{a} \int \frac{du}{u} \] Now we can solve the integrals: \[ \frac{m}{a}\ln|u| = -|v|t + C \] Substitute back for u: \[ \frac{m}{a}\ln(1 + |v|) = -|v|t + C \]

Numerical solution

To solve this equation numerically and find the bifurcation value \(a_0\), we can use numerical methods such as the Euler method or the Runge-Kutta method. By finding the limiting value of s(t) as \(t \to \infty\) for different values of a, we can identify the bifurcation value \(a_0\) at which the behavior of the system changes. Given the hint to refer to Exercise 14, you could proceed as follows:

Normalization and reference to Exercise 14

Divide by m and assume \(m/a = 1/x_0\) (similar to normalization). Take the limit as \(t \to \infty\) for the integral equation: \[ \ln(1 + |v|) = -x_0 s(t) + C, \] where \(s(t) = \int v\,dt\) is the displacement. Following the procedure of Exercise 14, find \(a_0\) such that \(\lim_{t \to \infty} s(t) = \infty\) if \(a \leq a_0\) and \(\lim_{t \to \infty} s(t)\) exists (finite) if \(a > a_0\). Once the bifurcation value \(a_0\) is found, we can also determine the limiting value of \(s(t)\) in the case where \(a > a_0\). By following these steps, one can set up and solve the given exercise using separation of variables and numerical methods.

Key concepts

Newton's Second Law

Newton's Second Law helps us understand how forces affect motion. In this exercise, it's crucial to form the equation of motion for the object moving through a resistive medium. The law states that the sum of forces on an object is equal to the mass of the object times its acceleration.

This exercise involves a resistive force proportional to the object's velocity. The resistive force is given by the formula \(a|v|/(1+|v|)\), highlighting the dependence on velocity \(v\). According to Newton's Second Law, the change in speed is dictated by this force. Thus, the movement's governing equation becomes:

\[ m \frac{dv}{dt} = -a \frac{|v|}{1 + |v|} \].

This equation models how the object's speed changes over time, incorporating both its mass \(m\) and the resistive force.

Numerical Methods

In situations where an analytical solution might be cumbersome or impossible, we resort to numerical methods. These methods are powerful tools for approximating solutions to differential equations.

Common methods include the Euler method and the Runge-Kutta method, used to approximate solutions by iterating step by step over small time intervals. By choosing different values of the resistive force constant \(a\), we can observe how the behavior of the system changes.

Numerical methods help confirm the results we get analytically. They allow us to explore how different parameters affect the solution, especially over limits that could not be solved directly. This suggests the numerical approach as an alternative to manually integrate and solve complex formulas.

Bifurcation Value

The concept of a bifurcation is threshold-based, where a small change in the parameter \(a\) results in a qualitative change in the system's behavior. In this exercise, a bifurcation value \(a_0\) determines whether the object will continue moving indefinitely or reach a stable speed.

Finding this critical \(a_0\) is key, as it tells us the precise point where the system's behavior changes from finite movement to infinite displacement.

By trying different assumptions through numerical calculations and plotting the solutions, we can identify \(a_0\) accurately. This value is significant since it marks the boundary of two distinct action types: forever moving objects versus those that settle into a stretch of calmness.

Separation of Variables

Separation of Variables is a fundamental technique to solve the differential equation by rearranging it. For this method, we isolate all terms involving the speed \(v\) on one side of the equation and the time \(t\) on the other side.

Consider the differential equation derived from Newton's law:
\[ \frac{m}{a} \frac{dv}{1 + |v|} = - |v| dt \].

By separating the variables and integrating each side, we solve the integrals for \(v\) and \(t\). The technique is to let one side reflect the resistive effects and the other the domain of time.

Once separated, we address each side of the equation by integrating, which leads us to expressions based on the variable changes. This step establishes the foundation for finding numerical or even analytical solutions if feasible.