A 5-m-diameter balloon and the gas within it have a mass of 80 kg. Determine its terminal veloc

step 1 At terminal speed, net downward force equals drag. The diameter is 5 meters, so the radius is 2

A 5-m-diameter balloon and the gas within it have a mass of...

Key Concepts

Air Density

Air density is a key parameter in problems involving objects moving through air, as it affects the magnitude of the drag force. At a given air temperature, such as 20°C, the density of air determines how many air molecules are present to interact with the object. A higher air density results in a larger drag force for a given velocity, which can lower the terminal velocity of the descending object.

Force Equilibrium

Force equilibrium in the context of terminal velocity refers to the condition where all forces on a body sum to zero, meaning there is no net acceleration. For a descending object, this equilibrium is achieved when the sum of the downward gravitational (or effective) force and any other forces, such as buoyant force, is balanced by the upward drag force, resulting in a steady descent velocity.

Cross-Sectional Area

The cross-sectional area is an important geometric parameter in drag calculations. For a spherical object, it is determined from its diameter using the formula A = ?*(d/2)^2. This area represents the size of the object facing the direction of motion and directly influences the magnitude of the drag force experienced by the object as it moves through the fluid.

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the net force acting on it becomes zero. This occurs when the downward gravitational force (adjusted for buoyancy, if applicable) is exactly balanced by the upward drag force. In this state, the object no longer accelerates, and its velocity remains steady as it descends through a fluid such as air.

Drag Force

Drag force is the resistance force experienced by an object moving through a fluid. It is typically modeled by the equation F_drag = (1/2) * C_d * ? * A * v^2, where C_d is the drag coefficient reflecting the object's shape, ? is the fluid density, A is the cross-sectional area perpendicular to the flow, and v is the velocity. This quadratic dependence on velocity means that small increases in speed can lead to large increases in drag force.

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