Would a chess board fall faster than the chess pieces?

Question

So if they were both falling in air, would the chess board fall faster since it's heavier?

Answer 1

Without air, so assuming the only affecting force is gravity, they will fall exactly equally fast! This was Galileo Galilei's discovery.

Sure, the heavier object (higher mass $m$) does indeed feel a larger downwards gravitational force. We can calculate it as its weight $w$, where $g$ is the gravitational acceleration it falls with:

$$w=mg.$$

But this larger force has a tougher time accelerating the object due to its larger mass. Newton described that these two effects actually turn out to exactly cancel each other out: an object with, say, double the mass feels double the force $w$ from the formula above but it is at the same time also is double as "slow" in speeding up (accelerating, $a$). Newton's 2nd law describes this:

$$\sum F=ma\quad\Leftrightarrow \quad w=ma \quad\Leftrightarrow \quad mg=ma\quad\Leftrightarrow \quad g=a.$$

As is seen mathematically, the mass cancels out of the equation; the mass has no influence on the falling motion! In total it thus falls just as fast (accelerates just as much) as any other object.

All this is the case in ideal no-air scenarios when the only influencing force is gravity (the weight). When other forces interact as well, such as air drag and the force of moving winds, then the total force $\sum F$ in Newton's 2nd law is constituted by more than just $w$, and then the accelerations may differ. In the case of air drag $D$, a typically used formula is:

$$D=\frac12 C_d A \rho v^2,$$

where $A$ is the frontal area, $\rho$ the density of the air, $v$ the falling speed and $C_d$ a drag coefficient that depends on factors such as shape (a parachute has a huge value of $C_d$ compared to something that is smooth and aerodynamic), surface roughness and the like. To figure out whether the chess board or chess pieces fall faster, we must look for differences among their values of these parameters.

Firstly, we realise that the chess board angle of attack is important. Depending on how it is angled during the fall, both $A$ and $C_d$ will be quite different depending on such angle. Still, though, for very low-weight chess pieces, I will expect yet other forces such as wind strokes to have higher impact.

The answer cannot be accurately given from a theoretical perspective since it depends on a lot of practical factors in a given scenario. But now we have at least analysed a bit on which factors to look out for in an emperical experiment.