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 (a) it is moving up?

Verified step by step guidance

1. Identify the forces acting on the baseball: When the baseball is thrown upward, two main forces act on it - the gravitational force (weight) downward and the drag force upward. The drag force is given as proportional to the square of the velocity (υ^2).
2. Set up the equation for net force: The net force acting on the baseball can be expressed as F_net = mg - kv^2, where m is the mass of the baseball, g is the acceleration due to gravity, k is a constant of proportionality, and v is the velocity of the baseball.
3. Determine the terminal velocity: Terminal velocity occurs when the net force on the object is zero (F_net = 0). This gives mg = kv_t^2, where v_t is the terminal velocity. Solve for v_t to find v_t = \sqrt{\frac{mg}{k}}.
4. Calculate the velocity when it is half the terminal speed: When the speed is half the terminal speed, v = \frac{1}{2}v_t. Substitute v_t from the previous step to find v = \frac{1}{2}\sqrt{\frac{mg}{k}}.
5. Find the acceleration: Substitute v = \frac{1}{2}v_t into the net force equation and solve for the acceleration a. The net force equation becomes F_net = m(g - k(\frac{1}{2}v_t)^2). Simplify and solve for a to find the y-component of the ball's acceleration when moving up.

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. For a baseball thrown upward, terminal velocity occurs when the gravitational force is balanced by the drag force, resulting in zero net acceleration.

Drag Force

The drag force is a resistive force acting opposite to the direction of motion of an object moving 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, affecting its acceleration.

Acceleration due to Gravity

Acceleration due to gravity (g) is the acceleration experienced by an object due to the gravitational pull of the Earth, approximately 9.81 m/s² downward. When analyzing the motion of the baseball, this acceleration must be considered alongside the drag force to determine the net acceleration of the ball at different speeds.