[Physics] If two objects have same density but different size,which one will drop first

dragnewtonian-gravitynewtonian-mechanicsprojectile

The greater the mass, the higher the terminal velocity. But what if the object's size (area) also increase, will the air resistance cause the heavier object to become slower than the lighter object?

Best Answer

Weight is $w=mg$ and air drag $D=\frac12 \rho C A v^2$. (Here $g$ is the gravitational acceleration, $C$ the drag coefficient, $A$ the cross section area seen from the flying direction, and $v$ the speed.)

Let's put both unto Newton's 1st law, which counts when there is terminal velocity (no more acceleration):

$$D-w=0\\ \frac12 \rho C A v^2-mg=0\\ v=\sqrt{\frac{mg}{\frac12 \rho C A}}=\sqrt{\frac{Vg}{\frac12 C A}}$$

This is the terminal speed. We have inputted $m=\rho V$.

  • Here it clearly shows that density doesn't really make a difference. But size does. Size in the form of volume, because that means more matter.

So you cannot simply say that mass has influence; two different types of stones can have same size and volume but different masses.

  • Shape also has an influence. Because of the cross section area. That is why a parachute falls slower but a spear faster.

So, the answer to your question is that yes, size can have an influence, but you need to know more. You need to know shape as well.