Physics initially aimed for describing and predicting the movement of matter. What is the essential property of any material object? Its position in space. For a simple object, its position can be summarized with a point in space, determined by a position vector $\vec{r}$. A complete description would involve information about its orientation.
What determines whether our object is ‘simple’ enough or not: the context. A car is a big object, yet it is represented by a single point in a GPS.
Things change in time, so position is not enough to specify the state of an object. The change of position is what we call velocity. Mathematically, it is represented by its first derivative with respect to time.
$$ \vec v=\frac{d\vec r}{dt} $$ Similarly, we can investigate the change of velocity in time (the change of change), and this is what we call acceleration. Mathematically it’s given by the second derivative with respect to time. $$ \vec a=\frac{d\vec v}{dt}=\frac{d^2\vec r}{dt^2} $$One could continue defining rates of change ad infinitum. Nevertheless, it is not necessary to go beyond acceleration, why? because for a given isolated object, one can always find a framework in which velocity remains constant. This is the first Newton’s law. A constant velocity implies that the movement follows a straight-line or rectilinear motion (uniform motion).
#Symmetries and Conservation Laws
The fact that velocity doesn’t change for an isolated object is an example of a conservation law (conservation of linear momentum in this case). Conservation laws are general principles that state that if a system fulfills certain conditions, there exists a certain quantity that remains constant in time. The most important conservation laws are the conservation of linear momentum, the conservation of angular momentum and the conservation of energy.
Why are conservation laws important? Simply because in physics we try to describe what changes in a system through what does not change. It could not be otherwise!
There is a close relationship between conservation laws and symmetries (which is the reason why physicists love symmetries): Conservation of linear momentum is obtained when a system looks the same when we displace it linearly in space; conservation of angular momentum is obtained when a system looks the same when we rotate it around a certain axis; finally, conservation of energy is derived when a system is indifferent to when we start counting time.
Forces