Thursday, 16 April 2015

newtonian mechanics - What is terminal velocity?


What is terminal velocity? I've heard the term especially when the Discovery Channel is covering something about sky diving. Also, it is commonly known that HALO (Hi-Altitude, Lo-Opening) infantry reaches terminal velocity before their chutes open.


Can the terminal velocity be different for one individual weighing 180 pounds versus an individual weighing 250 pounds?



Answer



Terminal velocity is the (asymptotic) maximum velocity that you can reach during free-fall. If you imagine yourself falling in gravity, and ignore air resistance, you would fall with acceleration $g$, and your velocity would grow unbounded (well, until special relativity takes over). This effect is independent of your mass, since


$F = ma = mg \Rightarrow a = g$


Where terminal velocity arises is that air resistance is a velocity-dependent force acting against your free fall. If we had, for example, a drag force of $F_D=KAv^2$ ($K$ is just a constant to make all the units work out and depends on the properties of the fluid you're falling through, and $A$ is your cross-sectional area perpendicular to the direction of motion) then the terminal velocity is the velocity at which the forces cancel (i.e., no more acceleration, so the velocity becomes constant):



$F = 0 = mg - KAv_t^2 \Rightarrow v_t=\sqrt{mg/KA}$


So we see that a more massive object can in fact have a larger terminal velocity.


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