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Classical Mechanics

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Moderate

Drag Force: Quadratic Drag Equal Fall Times

CLMECH-1ISXWB

The quadratic model for the drag force is:

$$F_d= cv^2 $$

...where $F_d$ is the drag force, $c$ is the drag coefficient and $v$ is the speed.

Two objects $A$ and $B$ are exactly the same except that the mass of object $B$, is twice the mass of object $A$ (i.e. $m_B=2m_A$). With sufficiently high drag ($c$ sufficiently large), the objects are dropped from different heights such that they strike the ground simultaneously.

Assuming that both objects achieve terminal speed almost immediately after being dropped, what is the ratio of the starting heights for objects $A$ and $B$ such that they will strike the ground simultaneously? Specifically, what is:

$$\cfrac{h_B}{h_A}=?$$

Note: $h_B$ is the height from which object $B$ is dropped and $h_A$ is the height from which object $A$ is dropped. Assume that the buoyant force due to air can be ignored.