#AP Physics C: Mechanics - Resistive Forces Study Guide 🚀

Hey! Let's dive into resistive forces – they're not as scary as they sound, promise! This guide is designed to get you prepped for the exam, focusing on the key concepts and how they all connect. Let's get started!

#1. Motion with Resistive Forces

#1.1. What are Resistive Forces?

• Resistive forces are like the brakes on a moving object. They always oppose motion, slowing things down. Think of them as the universe saying, "Not so fast!" 🐌

• The faster an object moves, the stronger the resistive force becomes. It's like trying to run through water – the faster you go, the harder it pushes back.

• A common example is air resistance. It's often modeled by the equation: $$\vec{F}_{r} = -k \vec{v}$$, where:

  • $$\vec{F}_{r}$$ is the resistive force vector.
  • k is a positive constant that depends on the shape and size of the object and the properties of the medium (like air density).
  • $$\vec{v}$$ is the object's velocity vector.

#1.2. The Math Behind the Motion: Differential Equations

• To really understand how resistive forces affect motion, we use Newton's Second Law ($$\vec{F}_{net} = m\vec{a}$$). When a resistive force is present, this leads to a differential equation for velocity.

• We use separation of variables to solve these equations. This allows us to find the velocity by integrating with appropriate limits. It's like solving a puzzle where each piece (variable) has its place.

• Once we know the velocity, we can find the acceleration and position functions using initial conditions and calculus. It's like building a story, one step at a time.

• For resistive forces like $$\vec{F}_{r} = -k \vec{v}$$, the resulting motion is described by exponential functions. This means the velocity, position, and acceleration don't change linearly but rather approach asymptotes. 📈

• Asymptotes are the limits that these functions approach as time goes to infinity. They're like the finish line that the object gets closer and closer to but never quite reaches.

• As time approaches infinity:

  • The velocity approaches a constant value (terminal velocity).
  • The acceleration approaches zero. This makes sense because the net force also approaches zero.

• The time constant, $$\tau = \frac{m}{k}$$, tells us how quickly an object reaches terminal velocity. Think of it as the object's "response time" to the resistive force.

  • A larger mass or smaller k (weaker resistive force) means a longer time to reach terminal speed. It's like a bigger boat taking longer to slow down in the water.

#1.3. Terminal Velocity: The Maximum Speed

• Terminal velocity is the maximum speed an object reaches when the constant force (like gravity) and the resistive force (like air resistance) balance each other. It's like a tug-of-war where both sides pull with equal strength, resulting in no net movement. ⚖️

• Mathematically, terminal velocity is reached when $$\frac{dv}{dt} = 0$$, meaning the acceleration is zero. The object stops speeding up.

• Objects with higher terminal velocities will take longer to reach that speed. It's like a car with a higher top speed needing more time to get there.

• Skydivers are a great example! In a spread-eagle position, they have a lower terminal velocity than in a streamlined head-first dive. This is because their surface area is greater and thus they experience more air resistance.

• Heavier objects have greater terminal velocities because they have a larger gravitational force to balance against air resistance. It's why a bowling ball falls faster than a feather.

#2. Final Exam Focus

#2.1. High-Priority Topics

• Differential equations for velocity and their solutions.
• Understanding terminal velocity and how it's achieved.
• The concept of the time constant and its implications.
• Analyzing graphs of position, velocity, and acceleration for objects experiencing resistive forces.

#2.2. Common Question Types

• Multiple Choice: Expect questions that test your understanding of the concepts, such as:
  • Identifying the direction of resistive forces.
  • Calculating terminal velocity in specific scenarios.
  • Interpreting graphs of motion with resistive forces.
• Free Response: You'll likely see problems that require you to:
  • Set up and solve differential equations for velocity.
  • Derive expressions for terminal velocity.
  • Analyze motion with varying resistive forces.

#2.3. Last-Minute Tips

• Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.
• Common Pitfalls: Be careful with units and signs. Always double-check your calculations.
• Strategies:
  • Start by drawing a free-body diagram to identify all the forces acting on the object.
  • Write out the differential equation and then solve it step-by-step.
  • Check your answers to see if they make sense in the context of the problem.

#3. Practice Questions

Practice Question

#Multiple Choice Questions

1. A ball is thrown vertically upward in the presence of air resistance. Which of the following statements is true regarding the ball's acceleration? (A) The acceleration is always directed downward and has a constant magnitude. (B) The acceleration is always directed downward, but its magnitude decreases as the ball rises. (C) The acceleration is always directed downward, but its magnitude increases as the ball rises. (D) The acceleration is directed upward when the ball is rising and downward when the ball is falling. (E) The acceleration is directed downward when the ball is rising and upward when the ball is falling.

2. An object of mass m is dropped from rest and experiences a resistive force given by $$F_r = -bv$$, where b is a positive constant and v is the velocity of the object. What is the terminal velocity of the object? (A) $$\frac{b}{mg}$$ (B) $$\frac{mg}{b}$$ (C) $$\frac{m}{b}$$ (D) $$\frac{b}{m}$$ (E) $$\frac{g}{b}$$

3. A small object is released from rest in a viscous fluid. Which of the following graphs best represents the object's velocity as a function of time? (A) A straight line with a positive slope (B) A curve that approaches a constant value (C) A parabola opening downward (D) A parabola opening upward (E) A horizontal line

#Free Response Question

A small sphere of mass m is released from rest in a viscous fluid. The sphere experiences a resistive force given by $$F_r = -kv$$, where k is a positive constant and v is the velocity of the sphere. The sphere also experiences a constant gravitational force mg acting downward.

(a) Draw a free-body diagram of the sphere as it falls through the fluid.

(b) Write the differential equation that describes the motion of the sphere in terms of m, k, g, and v.

(c) Determine the terminal velocity of the sphere in terms of m, k, and g.

(d) Solve the differential equation to find the velocity of the sphere as a function of time. Assume the initial velocity of the sphere is zero.

(e) Sketch a graph of the sphere's velocity as a function of time, labeling the terminal velocity.

#Scoring Breakdown:

(a) Free-Body Diagram (2 points)

• 1 point for correctly drawing the gravitational force (mg) acting downward.
• 1 point for correctly drawing the resistive force (kv) acting upward.

(b) Differential Equation (2 points)

• 1 point for correctly applying Newton's second law: $$ma = mg - kv$$
• 1 point for expressing acceleration as the derivative of velocity: $$m\frac{dv}{dt} = mg - kv$$

(c) Terminal Velocity (2 points)

• 1 point for setting the acceleration to zero when terminal velocity is reached: $$0 = mg - kv_t$$
• 1 point for solving for the terminal velocity: $$v_t = \frac{mg}{k}$$

(d) Solution to the Differential Equation (4 points)

• 1 point for separating the variables: $$\frac{dv}{mg - kv} = \frac{dt}{m}$$
• 1 point for correctly integrating both sides: $$-\frac{1}{k}ln(mg - kv) = \frac{t}{m} + C$$
• 1 point for using the initial condition v(0) = 0 to find the constant C: $$C = -\frac{1}{k}ln(mg)$$
• 1 point for solving for v(t): $$v(t) = \frac{mg}{k}(1 - e^{-\frac{k}{m}t})$$

(e) Graph of Velocity vs. Time (2 points)

• 1 point for showing an increasing curve that approaches a constant value.
• 1 point for labeling the terminal velocity on the graph as $$v_t = \frac{mg}{k}$$

Alright, you've got this! Remember to stay calm, focus on the fundamentals, and trust your preparation. You're ready to ace that exam! 🌟