Calculate the diameter of a parachute (in the form of a...

Calculate the diameter of a parachute (in the form of a hemispherical shell) to be used for dropping a small object of mass $90 \mathrm{~kg}$ so that it touches the earth at a velocity no greater than $6 \mathrm{~m} \cdot \mathrm{s}^{-1}$. The drag coefficient for a hemispherical shell with its concave side upstream is approximately $1.32$ for $\operatorname{Re}>10^{3}$. (Air density $=1.22 \mathrm{~kg} \cdot \mathrm{m}^{-3}$.)

Key Concepts

Terminal Velocity

Terminal velocity is the constant speed achieved by an object when the net force acting on it is zero; that is, when the downward force of gravity is exactly balanced by the upward resistive forces, such as air drag. In parachute design, ensuring the terminal velocity does not exceed a certain limit is essential for safety and effective operation.

Drag Force

Drag force is the resistance force exerted by a fluid (in this case, air) when an object moves through it. It is typically modeled by the equation F_drag = ½ ? v² C_D A, where ? is the air density, v is the velocity, C_D is the drag coefficient, and A is the effective area. This equation is central to determining the size and shape of a parachute required to achieve a designated terminal velocity.

Drag Coefficient

The drag coefficient is a dimensionless number that quantifies the drag or resistance of an object in a fluid environment. It encapsulates the effects of shape, flow conditions (such as Reynolds number), and surface roughness. In the context of parachute design, using the appropriate drag coefficient for a hemispherical shell is crucial for accurate calculations of the necessary canopy size.

Parachute Geometry

The geometry of the parachute, especially being a hemispherical shell, directly influences its effective area and the way air flows around it. For a hemispherical parachute, the projected area (which is used in the drag force calculation) is related to its diameter. Understanding this relationship is key in determining the appropriate dimensions needed to achieve the desired terminal velocity.

Force Equilibrium

Force equilibrium in the context of parachuting refers to the balance between the gravitational force (the weight of the object) and the drag force generated by the parachute. By setting the gravitational force equal to the drag force, one can solve for the required parameters (such as the parachute’s diameter) that ensure the object will descend at the targeted velocity.

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