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. Identify the forces acting on the baseball: gravity (downward) and drag force (upward when the ball is moving up and downward when the ball is moving down). The drag force is given as proportional to the square of the velocity, i.e., $F_d = k v^2$, where $k$ is a constant.
2. Set up the equation of motion using Newton's second law, $F = ma$, where $F$ is the net force on the baseball, $m$ is the mass of the baseball, and $a$ is its acceleration. The net force will be the difference between gravitational force and drag force.
3. Express the gravitational force as $F_g = mg$. Since the drag force is proportional to the square of the velocity, and the velocity is half the terminal velocity, substitute $v = \frac{1}{2}v_t$ into the drag force equation, giving $F_d = k (\frac{1}{2}v_t)^2 = \frac{k v_t^2}{4}$.
4. For the upward motion, calculate the net force as $F_{net} = F_g - F_d = mg - \frac{k v_t^2}{4}$. Use the terminal velocity condition where the drag force equals the gravitational force at terminal velocity, $k v_t^2 = mg$, to simplify the expression for net force.
5. For the downward motion, the drag force acts in the same direction as gravity, so the net force becomes $F_{net} = F_g + F_d = mg + \frac{k v_t^2}{4}$. Again, use the terminal velocity condition to simplify the expression for net force. Finally, substitute these net forces into Newton's second law to find the acceleration in each case.

Key Concepts

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the force of gravity is balanced by the drag force acting on it. For a baseball thrown upward, the terminal speed is the maximum speed it can achieve as it falls back down, where the gravitational force equals the drag force. Understanding terminal velocity is crucial for analyzing the motion of the baseball as it ascends and descends.

Drag Force

The drag force is a resistive force that opposes the motion of an object through a fluid, such as air. In this scenario, the drag force is proportional to the square of the velocity (υ²), meaning it increases significantly as the speed of the baseball increases. This concept is essential for calculating the net forces acting on the baseball and determining its acceleration at different speeds.

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. When analyzing the baseball's motion, both the gravitational force and the drag force must be considered to find the net acceleration. This concept is fundamental for understanding how the ball's speed and direction change as it moves upward and downward.