Newton’s second law
The state of motion of an object is reflected in its velocity vector. Retaining the original state of motion means there is no change in the velocity vector: the object continues to move in the same direction with the same speed – this means the acceleration is zero. A change in the state of motion means that the magnitude or the direction of velocity changes, which is equivalent to a non-zero acceleration.
In order to effect a change in the state of motion of an object, another object must exert a force on it.
Force
Force is a push or pull upon an object resulting from the object's interaction with another object; an action that causes the object to change its state of motion.
Force is a vector quantity. Consider, for example, billiard balls hit by the cue: they will start to move in the direction of the push, which shows us that the effect of the force must be different if the force acts in a different direction.
The SI unit of force is the kilogramme metre per second squared, which is also called the newton:
\[\left[\mathbf{F}\right] = 1\,\mathrm{kg} \cdot \frac{\mathrm{m}}{\mathrm{s}^2} = 1\,\mathrm{N}.\]
Balanced and unbalanced forces
The presence of a force does not guarantee that the state of motion of the object will change. Consider, for example, a book placed on a table: it retains its state of motion, although two forces act upon it – gravity pulls it down and the table pushes it up. In this case, these forces balance each other, so there is no change in the state of motion of the book. By contrast, the state of motion of a book in free fall keeps changing as the force of gravity pulls it down and there is no other force that could balance that.
Newton's first law states that if there are no forces, or all forces are balanced, the object keeps its state of motion. By contrast, if there are unbalanced forces, the state of motion will change and the object will accelerate.
The greater the force acting upon objects with the same mass, the greater the acceleration, whilst the greater the mass of an object upon which a force acts, the less the acceleration. These relationships are summarised in Newton's second law.
Newton’s second law
The acceleration of an object is directly proportional to the force acting on it and inversely proportional to its mass. The direction of the acceleration is the same as the direction of the force.
\[\mathbf{F} = m \mathbf{a}\]