Why F = m * a? vs Why not F = m + a?

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The equation F = m * a is fundamentally rooted in the relationship between force, mass, and acceleration, emphasizing that these quantities cannot be simply added together like apples and oranges. The discussion highlights the importance of dimensional analysis, noting that combining mass and acceleration requires careful consideration of units to maintain mathematical validity. While alternative formulations could theoretically exist with constants, they would not hold the same empirical significance as Newton's second law. Ultimately, the validity of F = m * a is supported by experimental evidence rather than arbitrary mathematical manipulation. This underscores the principle that physical laws are derived from observed phenomena rather than mere mathematical convenience.
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Why F = m * a?

Why not F = m + a?
 
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Adding acceleration and mass is just like adding apples and oranges, it can't be done.

Look at the units involved and combine them in a manner which makes sense.
 
Of course, it would be (mathematically) possible to have something like F= C_1mz+ C_2a where C_1 and C_2 are constants with the correct units (C_1 would have to have units of "Newtons per kg" or "meters per second squared" and C_2 has units of "Newtons-seconds square per meter" or "kilograms".

But the only good Physics explanation as to why a specific formula is true is "that is what the experimental evidence shows".
 
From: http://en.wikipedia.org/wiki/Newton's_laws_of_motion

"Newton's original Latin reads:
Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.
This was translated quite closely in Motte's 1729 translation as:
Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd."
 

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