From net force to acceleration
The first law told us that a nonzero net force changes an object's motion. Newton's second law tells us exactly how much: the net force equals mass times acceleration. Push twice as hard and you accelerate twice as fast; load twice the mass and, for the same push, you accelerate half as fast.
Newton's second law: the net force on a body equals its mass times its acceleration; the acceleration points along the net force.
Read the equation as an arrow statement: the acceleration lies in the same direction as the net force, and its size is proportional to the force and inversely proportional to the mass. Zero net force gives zero acceleration — and we are right back at the first law, which is just the special case \vec{a}=0.
The newton, and getting units right
Force needs a unit, and the second law defines it. One newton (N) is exactly the net force that gives a one-kilogram mass an acceleration of one metre per second squared. It is not a fundamental unit but a combination of kilogram, metre and second.
The newton expressed in SI base units.
It is a vector law: work one axis at a time
Because \vec{F}_{\text{net}} = m\vec{a} is a vector equation, it is really two (or three) equations at once — one for each direction. In practice you pick convenient axes, then demand that the net force along each axis equals mass times the acceleration along that same axis.
The second law resolved into independent component equations along perpendicular axes.
This is the whole strategy of dynamics: turn a tangle of arrows into two clean scalar equations. A projectile, for instance, has a_x = 0 and a_y = -g; a block sliding on level ground has a_y = 0 while all the action lives in the x-equation. Choosing the right axes turns a hard problem into an easy one.
Weight versus mass — do not confuse them
Mass is how much matter — how much inertia — an object has. It is measured in kilograms and is the same on Earth, on the Moon, or in deep space. Weight is something different: it is the gravitational force on that mass, measured in newtons, and it is just the second law applied to free fall.
Weight equals mass times the local gravitational acceleration g (about 9.8 m/s^2 near Earth's surface).
Worked example: pushing a crate
A 20 kg crate sits on a smooth (frictionless) floor. You push it horizontally with a steady 60 N. What is its acceleration? Step 1 — isolate the crate and identify horizontal forces: just the 60 N push (gravity and the floor's push cancel vertically). Step 2 — apply the second law along the direction of motion.
The crate accelerates at 3 m/s^2 in the direction of the push.
That is the entire method in miniature: draw the object alone, sum the forces on it, divide the net force by the mass. In the real world, though, the floor is never perfectly smooth — friction would push back and reduce that acceleration. To handle that honestly we first need a careful way to catalogue every force on an object: the free-body diagram, and Newton's third law that generates force pairs. Those come next.