When, considering subsequent instants, the ratiox /t is no longer constant, and therefore the point P travels different spaces in equal times, the motion is defined as non-uniform rectilinear and the speed will consequently depend on the chosen instants of time; in this case, instantaneous speed is used.
The instantaneous speed
The instantaneous velocity of the point P is that possessed by the moving body at the instant in which it occupies exactly the position defined by the coordinate x1. To define the instantaneous speed, a mathematical operation must be performed, indicated as the limit of the incremental ratio, consisting in considering smaller and smaller time intervalst , making the x coordinate approach2at x1until the space traveled by the point along the straight line is canceled. The instantaneous velocity of P at point x1it will therefore be the one resulting from the ratiox /t , whent tends to assume values ever closer to zero. Since, as reported by the mathematical analysis, this operation defines the derivative of the hourly law s = s ( t ), which provides the space traveled s as a function of the time t used, it is therefore possible to also resort to the differential calculation for the definition of the different kinematic quantities.
acceleration
In the case of a motion at non-constant speed, a second vector quantity called acceleration can also be introduced, a , always having direction along the straight path and towards in the direction of motion, whose module allows to describe the variation of speed over time. As in the case of uniform motions, it is possible to initially define an average acceleration, am:
which represents the relationship between the change in the speed of a body and the time interval in which this change took place. If this ratio remains constant over time, we can speak of a uniformly accelerated rectilinear motion; otherwise, the concept of instantaneous acceleration can also be introduced, as a limit of the incremental ratiov /t .
Acceleration has the size of a space divided by a time squared. In the International System, the unit of measurement of acceleration is that of a body that varies its speed by 1 m / s in the interval of 1 s, expressed by the symbol m / s 2 , which reads “meter per second squared “.
The hourly law of uniformly accelerated rectilinear motion
The hourly law for the simplest type of non-uniform rectilinear motion, that is the uniformly accelerated one (at constant acceleration), is obtained considering a point P which, in a time interval of t seconds, moves along a straight line, varying its own speed from a value v0= 0 at the initial instant t = 0 up to a value vtat the final instant t .
As previously stated on the average value of a series of measurements, the point P will have an average speed:
The case in question can therefore be traced back to that of a point that moves with uniform rectilinear motion, with a constant speed equal to half the final speed vt. Applying the relative law of motion, we obtain the relation:
which, combined with:
allows to obtain the hourly law of uniformly accelerated rectilinear motion:
In the case of a point P which, at the initial instant t = 0, has a velocity v0 not nothing, the relationships seen so far will take the form:
But, being:
it turns out that:
from which:
namely:
Finally, considering a point P which occupies an initial position s0 different from the origin, the more general hourly law of uniformly accelerated rectilinear motion assumes the definitive form:
In a Cartesian plane, the hourly law of uniformly accelerated rectilinear motion is represented by a second degree curve, called a parabola.
