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7. Friction, Inclines, Systems
Kinetic Friction
Problem 40b
Textbook Question
You throw a baseball straight upward. The drag force is proportional to v2. In terms of g, what is the y-component of the ball's acceleration when the ball's speed is half its terminal speed and it is moving back down?

1
Step 1: Understand the forces acting on the baseball. When the ball is moving upward or downward, two forces act on it: gravity (mg, acting downward) and the drag force (proportional to υ², acting opposite to the direction of motion). Terminal velocity occurs when the drag force equals the gravitational force, meaning the net acceleration is zero.
Step 2: Express the drag force mathematically. The drag force can be written as F_drag = kυ², where k is a proportionality constant and υ is the velocity of the ball. At terminal velocity (υ_terminal), the drag force equals the gravitational force: kυ_terminal² = mg.
Step 3: Relate the drag force to the current velocity. When the ball's speed is half its terminal velocity (υ = 0.5υ_terminal), the drag force becomes F_drag = k(0.5υ_terminal)² = k(υ_terminal²/4). Substitute kυ_terminal² = mg into this expression to find F_drag = mg/4.
Step 4: Determine the net force and acceleration when the ball is moving upward. The net force is the sum of the forces acting on the ball: F_net = -mg (gravity, downward) - F_drag (drag, downward). Substituting F_drag = mg/4, the net force becomes F_net = -mg - mg/4 = -(5/4)mg. Use Newton's second law (F_net = ma) to find the acceleration: a = F_net/m = -(5/4)g.
Step 5: Determine the net force and acceleration when the ball is moving downward. In this case, gravity acts downward (positive direction), and the drag force acts upward (negative direction). The net force is F_net = mg (gravity, downward) - F_drag (drag, upward). Substituting F_drag = mg/4, the net force becomes F_net = mg - mg/4 = (3/4)mg. Use Newton's second law to find the acceleration: a = F_net/m = (3/4)g.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Terminal Velocity
Terminal velocity is the constant speed an object reaches when the drag force acting on it equals the gravitational force, resulting in zero net acceleration. For a baseball thrown upward, this speed is influenced by its mass, shape, and the density of the air. Understanding terminal velocity is crucial for analyzing the motion of the baseball as it ascends and descends.
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Drag Force
The drag force is a resistive force that opposes the motion of an object through a fluid, such as air. It is often modeled as being proportional to the square of the object's velocity (υ²). This concept is essential for determining how the baseball's speed affects its acceleration, especially when considering forces acting on it during its flight.
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Acceleration due to Gravity
Acceleration due to gravity (g) is the rate at which an object accelerates towards the Earth when in free fall, approximately 9.81 m/s². This acceleration acts downward and is a key factor in the motion of the baseball. When analyzing the ball's motion, both the gravitational force and the drag force must be considered to determine the net acceleration at various speeds.
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