Chapter 2: Problem 40
Consider an object that is coasting horizontally (positive \(x\) direction) subject to a drag force \(f=-b v-c v^{2} .\) Write down Newton's second law for this object and solve for \(v\) by separating variables. Sketch the behavior of \(v\) as a function of \(t .\) Explain the time dependence for \(t\) large. (Which force term is dominant when \(t\) is large?)
Short Answer
Step by step solution
Write Down Newton's Second Law
Separate Variables
Integrate Both Sides
Solve for Velocity \( v(t) \)
Determine Long-Time Behavior
Sketch \( v(t) \) and Analyze Dominant Term
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Drag Force
- **Linear Drag:** The term \( -b v \) represents linear drag. It is proportional to the velocity \( v \) and comes into play at lower speeds or for streamlined objects.
- **Quadratic Drag:** The \( -c v^2 \) term denotes quadratic drag. This form of resistance becomes significant at higher speeds, where the object's shape and the density of the fluid highly influence the motion.
Separation of Variables
To separate variables, we rearrange the equation in a way that one variable is on each side. This looks like: \[ \frac{dv}{b v + c v^2} = -\frac{1}{m} dt \].
This method allows us to tackle the problem by integrating each side independently. The left side is integrated with respect to \( v \) and the right side with respect to \( t \). This process unravels the dynamics of how velocity evolves over time under the influence of drag forces.
Asymptotic Behavior
\[ v = -\frac{b}{c} \]
- This result indicates that the velocity approaches a constant value over time, determined purely by the parameters \( b \) and \( c \).
- The constant velocity implies that quadratic drag \( -c v^2 \) becomes the dominant force as time progresses, stabilizing the motion.
Differential Equation Solutions
Using partial fraction decomposition simplifies the integral of the left side. After integration, we arrive at:
\[ \frac{1}{b} \ln|b + c v| = -\frac{t}{m} + C \], where \( C \) is a constant determined by initial conditions.
Solving for \( v \), we get:
\[ v(t) = \frac{K e^{-\frac{bt}{m}} - b}{c} \]
Here, \( K \) integrates initial conditions, ensuring the solution is specific to the scenario's starting details. Understanding the procedure for solving these equations equips students to address a variety of problems involving changing rates, which are common in both theoretical and practical physics contexts.