# If two objects are released from a certain height both will reach at the same time.but according to fluid dynamics (terminal velocity) the denser will reach first.does it mean that the concept of terminal velocity is wrong??

Jun 14, 2018

No! Both statements are right - but within their respective contexts.

#### Explanation:

The statement that both objects will reach (the ground) together assumes that both objects are subjected to only gravitational force. In this case, the two bodies will accelerate equally.

On the other hand, if the objects fall through a fluid which offers appreciable resistance, it is true that the their accelerations will differ and one will reach the ground ahead of the other.

Assuming that the motion is sufficiently slow throughout, we can use Stokes' law to find out the drag force and hence write down the equation of motion for an object (assumed to be a sphere of radius $a$) as

$m a = m g - 6 \pi \eta a v \implies$

$a = g - 6 \pi \eta \frac{a}{m} v$

Using $m = \frac{4}{3} \pi {a}^{3} \rho$ we get

$a = g - \frac{9}{2} \frac{\eta}{\rho {a}^{2}} v$

Note that this means that the terminal velocity (achieved when $a \to 0$) is actually proportional to $\rho {a}^{2}$ - not just $\rho$. So, the statement that denser bodies have bigger terminal velocities is true only as long as they have the same radius. A denser, but smaller sphere, may have a smaller terminal velocity!