91Ó°ÊÓ

Falling Baseball. You drop a bascball from the roof of a tall building. As the ball falls, the air exerts a drag force proportional to the square of the ball's speed \(\left(f=D v^{2}\right) .\) (a) In a diagram, show the direction of motion and indicate, with the aid of vectors, all the forces acting on the ball. (b) Apply Newton's second law and infer from the resulting equation the general properties of the motion. (c) Show that the ball acquires a terminal speed that is as given in Eq. (5.13). (d) Derive the equation for the speed at any time. (Note: $$ \int \frac{d x}{a^{2}-x^{2}}=\frac{1}{a} \operatorname{arctanh}\left(\frac{x}{a}\right) $$ where $$ \tanh (x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=\frac{e^{2 x}-1}{e^{2 x}+1} $$ defines the hyperbolic tangent.)

Step by step solution

01

Diagram of Forces

First, identify the forces acting on the baseball as it falls. The two main forces are gravity, which acts downward, denoted by \( F_g = mg \) where \( m \) is the mass and \( g \) is the acceleration due to gravity, and the drag force, which acts upward and is given by \( F_d = Dv^2 \), where \( D \) is the drag coefficient and \( v \) is the velocity. The vector diagram should show these forces with the direction of motion being downward.

02

Apply Newton's Second Law

Write Newton's second law for the falling baseball. The net force \( F_{net} \) is the difference between the gravitational force and the drag force: \( F_{net} = mg - Dv^2 \). According to Newton's second law, \( F_{net} = ma = m \frac{dv}{dt} \). Therefore, \( m \frac{dv}{dt} = mg - Dv^2 \). This equation will help infer the characteristics of motion such as acceleration, velocity changes, and terminal speed.

03

Terminal Speed Derivation

At terminal speed, the net force is zero because the gravitational force is balanced by the drag force. Thus, set \( mg = Dv_t^2 \), where \( v_t \) is the terminal speed. Solving for \( v_t \), you get \( v_t = \sqrt{\frac{mg}{D}} \). This result confirms that the ball reaches a constant speed when falling, known as terminal speed.

04

Solve Differential Equation for Velocity

Using the differential equation \( m \frac{dv}{dt} = mg - Dv^2 \), separate variables and integrate both sides. Rearrange to form: \( \frac{dv}{g - (D/m)v^2} = dt \). For integration, use: \( \int \frac{dv}{a^2 - x^2} = \frac{1}{a} \operatorname{arctanh} \left(\frac{x}{a}\right) \) to find a solution that describes \( v(t) \). This should yield \[ v(t) = v_t \tanh \left( \frac{g}{v_t} \frac{D}{m} t \right) \], representing the velocity of the ball as a function of time, approaching terminal velocity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Drag Force

When an object moves through a fluid like air, it experiences a resistance force known as drag force. For our falling baseball, this force opposes the motion and is proportional to the square of its speed, expressed as \( F_d = Dv^2 \). Here, \( D \) is the drag coefficient, which depends on factors like the shape and cross-sectional area of the ball, as well as the density of air. Understanding drag force is crucial because it significantly affects the motion of objects moving at high speeds through a fluid. As speed increases, the drag force becomes stronger, leading to a balance with the gravitational force when terminal velocity is reached.

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the force of gravity pulling it downwards is exactly balanced by the drag force pushing upwards. At this point, the net force acting on the object is zero, causing it to stop accelerating. For the baseball in our problem, terminal velocity \( v_t \) can be derived by setting the gravitational force equal to the drag force: \( mg = Dv_t^2 \). Solving for \( v_t \) gives us \( v_t = \sqrt{\frac{mg}{D}} \). This formula shows that terminal velocity depends on the mass of the object, the gravitational force, and the drag coefficient. This balance leads the baseball to fall steadily at this speed without increasing or decreasing speed.

Differential Equations

To describe the changing velocity of the falling baseball over time, we use a differential equation derived from Newton's second law. This equation is \( m \frac{dv}{dt} = mg - Dv^2 \), which shows the balance between gravitational force and drag force as the ball falls.This equation is a tool to understand how acceleration and velocity change over time. To solve it, we separate variables and integrate, leading to the relationship \[ v(t) = v_t \tanh \left( \frac{g}{v_t} \frac{D}{m} t \right) \]. This solution provides a function for velocity in terms of time, illustrating how velocity approaches terminal velocity as time progresses. Differential equations like this are valuable in modeling real-world phenomena where variables change continuously.

Gravity

Gravity is the fundamental force driving the fall of the baseball. It is a constant pull towards the center of the Earth and provides the downward force in our scenario. Calculated as \( F_g = mg \), gravity is proportional to the mass of the baseball and the acceleration due to gravity, \( g \), which is approximately \( 9.8 \text{m/s}^2 \) on Earth.In the context of the exercise, gravity works in opposition to the drag force. Understanding how gravity interacts with other forces, like drag, helps us comprehend why objects reach terminal velocity and how they move through fluids. This balance of forces is a practical application of Newton's second law, illustrating the predictability in the motion of an object.