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Is there a difference between "No Air Resistance" and "Constant Air Resistance''?

Like for example if we have two identical objects with different masses and we drop them from the same height.

Will their motions be the same if there is no air resistance and if there is an air resistance assuming it is constant on the two objects?

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  • $\begingroup$ Constant wrt. to what? Time? Position? Shape? $\endgroup$ – Qmechanic Oct 30 '14 at 19:45
  • $\begingroup$ I just added some extra info. ^ $\endgroup$ – Maher Oct 30 '14 at 19:52
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    $\begingroup$ I presume when you say 'identical' you mean the same size and shape? No air resistance is the special case of (constant) zero air resistance... $\endgroup$ – Jon Custer Oct 30 '14 at 20:42
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If the two objects are identical in shape the initial drag force on both are equal since drag is a function of the shape and velocity. However, since the two objects have different mass the resultant deceleration rate is different, and the heavier object will decelerate slower than the lighter object, and this deceleration rate difference will in effect change the drag force (since drag force is proportional to velocity squared). In this sense, the two objects will likely not hit the ground at exactly the same time. The kinematic equation governing motion of the object is:

$d^2x/dt^2 = g - 1/2\rho AC_d(dx/dt)^2/m_{object}$

Note that $C_d$ is a function of the Reynolds number which will vary with velocity but depending on your velocity range and shape it may be treated as a constant. Solving the above equation will give x(t) and given the height you drop from you can find the time it takes for each object to impact the ground.

In a vacuum both objects will hit the ground at the same time.

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Well if you define "constant" right, they'll drop at the same rate. However, while their relative motions are the same, it's still different from dropping them in a vacuum (they'll both fall faster in this case).

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