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Question

1. Can Newton's first law of motion be considered merely as a special case of second law of motion with $a=0$? If so, the first law really needed?

2. Why does the acceleration of a freely falling body not depend on the mass of the object?

Answer 1

1. Yes, Newton’s first law is a special case of his second law. It is not logically necessary, but it is useful to explain the special case of what happens with no applied force before proceeding to the general case. Newton probably started with his first law in order to clearly distinguish his theory from Aristotle’s theory of motion, in which the “natural state” of all objects is to be at rest.

2. The simple answer is because the weight of an object is proportional to its mass. Newton’s second law tells us that $W =ma$ where $W$ is the weight of the object, but if $W=mg$ then $m$ appears on both sides of the equation of motion and cancels out. But the really interesting question is why gravitational mass (the $m$ in $W=mg$) should be always the same as inertial mass (the $m$ in $F=ma$). It was thinking about this question that led Einstein to discover the theory of general relativity.