You throw a baseball straight upward. The drag force is proportional to υ2. 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 (b) It is moving back down?

Verified step by step guidance

1. Understand the forces acting on the baseball: When the baseball is thrown upward, it experiences gravitational force (mg downward) and a drag force proportional to the square of its velocity (F_drag = kυ² upward). Terminal velocity is reached when the drag force equals the gravitational force.
2. Define terminal velocity: Terminal velocity (υ_t) occurs when the drag force equals the gravitational force, i.e., kυ_t² = mg. From this, you can express the drag coefficient k in terms of g and υ_t: k = mg/υ_t².
3. Determine the drag force when the speed is half the terminal speed: If the speed υ is half the terminal speed, then υ = υ_t/2. Substitute this into the drag force equation: F_drag = k(υ_t/2)² = kυ_t²/4.
4. Calculate the net force when the ball is moving upward: The net force is the difference between the gravitational force and the drag force. F_net = mg - kυ_t²/4. Use Newton's second law to find the acceleration: a = F_net/m = g - (g/4) = 3g/4.
5. Calculate the net force when the ball is moving downward: When moving downward, both the gravitational force and drag force act in the same direction. F_net = mg + kυ_t²/4. Use Newton's second law to find the acceleration: a = F_net/m = g + (g/4) = 5g/4.

Key Concepts

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration. It occurs when the drag force equals the gravitational force, resulting in zero net acceleration. Understanding terminal velocity is crucial for analyzing the motion of objects with drag forces.

Drag Force

Drag force is a resistance force caused by the motion of a body through a fluid, such as air. It is often proportional to the square of the velocity (υ²) for high-speed objects. This concept is essential for calculating the net force acting on the baseball, which affects its acceleration as it moves upward and downward.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma). This law is fundamental for determining the y-component of the ball's acceleration by considering both gravitational and drag forces acting on the baseball.