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adam12349 t1_j8jvlcq wrote

Ok lets dive into this so with no air resistance things accelerate until they reach the ground.

Lets have to objects with mass m and M.

The force of gravity F=m×g

So m×a=m×g or M×a=m×g.

Mass factors out a=g in most cases. Cool that was demonstrated on the Moon with a hammer and a feather.

For Newtonian gravity F=G×M×m×1/r² where M is the mass of the planet.

So lets use m for our object and see if it drops out.

ma=GM/r² × m

a=GM/r² Great.

Now lets look into air resistance and terminal velocity. An object reaches terminal velocity when the sum of Fg and Fr (for resistance) is 0. So the forces are in equilibrium.

Fg=Fr.

Fr=½×q×C×A×v² where A is the surface area of the object C is a factor for shape q is the density of the medium and v is the velocity. We can say taht q and C are constant so lets combine the into a factor b. (With the ½.) We will use A and get a formula for v the terminal velocity.

This all leads to the square-cube law if we assume our different objects have the same density.

Fr=bv²A

Lets look at a solid ball with density q. Its volume is V=4/3×pi×r³. So from q=m/V we get

m=qV=q×4/3pi×r³.

Now A=4pi×r². And with that lets plug that into Fr=Fg=mg.

q×4/3×pi×r³×g=bv²×4pi×r²

lets rearrange

qrg=bv²×3

So now we get

⅓×qg/b × r=v² that first bit is a constant so lets call it k

k×r=v²

(k×r)^½ = v

So with same density balls the larger falls faster. If you increase the radius by a factor of 4, v increases by a factor of 2.

The m(r) function is simple we already have that

m=qV=q×4/3×pi×r³

So now for r(m)

r = (3/4×m/(q×pi))^⅓

So now lets plug it into the v² formula.

v²=k×(3/4×m/(q×pi))^⅓

v²=k × (3/4)^⅓ × (1/q×pi)^⅓ × m^⅓. Bunch of constants and m so lets combine them into K giving us.

v²=K × m^⅓

v=K^½ × m^⅕ Lets call K^½ = C

v=C×m^⅕

So as mass increases so does v but that was obvious from the v(r) formula.

To know the value of C you need the shape factor for our sphere its 0.47 the density of the fluid (air) g~9.81m/s² pi is 5 of course and the density of our material. Of course for different shapes the resulting functions can look different but the square-cube law will apply. More mass height terminal velocity. But the results may be very similar as any shape is appropriately a sphere.

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