F = ma: the second law

The first law says velocity changes only when a net force acts. The second law finishes the sentence: it says how much.

The law

$F_{net} = ma$

Three quantities, one relationship, read best as a = F/m: acceleration is what the net force buys, per kilogram of mass.

• Double the force on the same mass → double the acceleration.
• Same force on double the mass → half the acceleration.
• Zero net force → zero acceleration: the first law, now as arithmetic.

Units: one newton is the force that gives 1 kg an acceleration of 1 m/s². About the weight of an apple — Newton would have appreciated that.

Net force, not "the" force

The F in the law is the sum of every force acting. A 1000 kg car whose engine pushes 3000 N forward against 1000 N of drag has a net force of 2000 N, so a = 2 m/s². Forget the drag and every answer comes out wrong. The bookkeeping tool is the free-body diagram: draw the object alone, draw every force on it, add them up.

Weight vs mass — and why everything falls together

Your mass m is how much matter you are (kilograms, same on the Moon). Your weight is the gravitational force on that mass: W = mg. Now run free fall through the second law:

$a = \frac{F}{m} = \frac{mg}{m} = g$

The m cancels. Twice the mass means twice the pulling force and twice the sluggishness — so every object gets the same g. The hammer-and-feather tie from module 2 isn't a coincidence; it's this cancellation.

Why this matters

F = ma is the engine of mechanics: give me the forces and I'll tell you the motion. It's how we knew the constant-acceleration toolkit would apply to gravity — constant force, constant mass, therefore constant a.