📚 Topic Summary
When an object falls through the air, it experiences two main forces: gravity, which pulls it downwards, and air resistance, which opposes its motion. The force of gravity is usually modeled as $mg$, where $m$ is the mass of the object and $g$ is the acceleration due to gravity (approximately $9.8 m/s^2$). Air resistance, however, is more complex and can be modeled in different ways. One common model is to assume that air resistance is proportional to the velocity of the object, i.e., $kv$, where $k$ is a constant that depends on the shape and size of the object and the density of the air. Another model assumes that air resistance is proportional to the square of the velocity, i.e., $kv^2$.
The differential equation that describes the motion of a falling object is obtained by applying Newton's second law, which states that the sum of the forces acting on an object is equal to its mass times its acceleration ($F = ma$). Since acceleration is the derivative of velocity with respect to time ($a = \frac{dv}{dt}$), we can write the differential equation as $m\frac{dv}{dt} = mg - kv$ (for air resistance proportional to velocity) or $m\frac{dv}{dt} = mg - kv^2$ (for air resistance proportional to the square of velocity). Solving these differential equations allows us to determine the velocity of the falling object as a function of time.
🧪 Part A: Vocabulary
Match the term with its correct definition:
- Terminal Velocity
- Air Resistance
- Differential Equation
- Gravity
- Initial Condition
Definitions:
- A condition that is given at the initial time.
- The force that attracts a body toward the center of the earth, or toward any other physical body having mass.
- A mathematical equation that relates a function with its derivatives.
- The constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration.
- The force that opposes the motion of an object through the air.
Match the number to the correct term.
📊 Part B: Fill in the Blanks
When an object falls, it accelerates due to ________. However, as its speed increases, ________ also increases, acting in the ________ direction. Eventually, these forces balance, and the object reaches ________, where its acceleration becomes ________.
🤔 Part C: Critical Thinking
Explain how the shape and size of a falling object affect its terminal velocity, considering the role of air resistance. Provide real-world examples.