## Do spherical objects with the same radius, but with diferent densities fall with the same speed and acceleration?

Question

Taking into account different masses and the square law of drag. Do they fall through the air at the same time or not?

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General Physics
2 years
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## Answer ( 1 )

Force of gravity is defined as:

Fg = G m₁ m₂ / r²,

where G is the gravitational constant,

m₁ and m₂ are the masses (in this case, the earth and the sphere),

and r is the distance between them.

Drag is defined as:

Fd = ½ ρ v² Cd A,

where ρ is the density of the fluid,

v is the velocity,

A is the cross sectional area,

and Cd is the drag coefficient.

Considering only these forces, Newton’s second law states:

∑F = ma

Fd – Fg = ma

(½ ρ v² Cd A) – (G m M / r²) = ma

a = (½ ρ v² Cd A / m) – (G M / r²)

So a larger m would result in a more negative acceleration (falls faster) than a smaller m.