Way-time law

In classical physics, a distance-time law describes the course of the movement of a mass point . It is valid for a certain movement in each case by giving the location of the mass point as a function of time. It thus represents the course of the movement of a body over time on its trajectory and is therefore also referred to as a time-location function . Given the external forces , it is the special solution of the equation of motion of the mass point determined by the initial conditions for location and velocity .

If the movement is determined from the outset by constraints on a certain line, such as the movement of a locomotive through the rails, the arc length along the track, which is then usually referred to as the path or route , is sufficient as a location specification . The zero point of the path can be freely selected. The movement can then be represented in a function graph called a time-location diagram . In all other cases, the time-location function gives the independent coordinates of the mass point relative to a freely chosen reference system at the given time and is therefore vector-valued .

The symbol for the value of the distance-time function is often , or something similar. This is to express that the place is a clear function of the time , which represents a free variable in the mathematical sense. Exactly one place is assigned to each point in time , where the mass point is currently located. The reverse is not true: A mass point can very well be in the same place at different times. The distance-time function is continuous , as the mass point cannot “jump” from one place to another without losing time. Expressed in mathematical terms: the distance that the mass point can cover tends to zero when the available time interval also tends to zero. Furthermore, the distance-time function can be differentiated once - at least in sections ; if the speed does not change abruptly, even twice. The first time derivative , often referred to as after Isaac Newton , is the instantaneous velocity . This function is also called the speed-time law or time-speed function . The second derivative gives the acceleration . ${\ displaystyle {\ vec {r}} (t)}$ ${\ displaystyle {\ vec {X}} (t)}$ ${\ displaystyle {\ vec {r}}}$ ${\ displaystyle t}$ ${\ displaystyle {\ dot {\ vec {r}}} (t)}$ ${\ displaystyle {\ vec {v}} (t) = {\ dot {\ vec {r}}} (t)}$ ${\ displaystyle {\ vec {a}} (t) = {\ dot {\ vec {v}}} (t) = {\ ddot {\ vec {r}}} (t)}$

The representation of the coordinates of the location depends on the selected coordinate system . For a movement in a plane, for example, is in a two-dimensional Cartesian coordinate system , or alternatively in polar coordinates . The number of components of is equal to the number of dimensions of the space in which the movement takes place. ${\ displaystyle {\ vec {r}} (t) = (x (t), y (t))}$ ${\ displaystyle {\ vec {r}} (t) = (r (t), \ varphi (t))}$ ${\ displaystyle {\ vec {r}} (t)}$

Examples

The following examples describe idealized, simplified processes. All movements start at the time at the starting point indicated by . ${\ displaystyle t = 0}$ ${\ displaystyle {\ vec {r}} _ {0}}$

At a standstill, the position does not depend on the time and the mass point remains at the starting point forever : ${\ displaystyle {\ vec {r}} _ {0}}$

{\ displaystyle {\ vec {r}} (t) = {\ vec {r}} _ {0} = \ mathrm {const.}}

Uniform rectilinear movement with speed : ${\ displaystyle {\ vec {v}}}$

{\ displaystyle {\ vec {r}} (t) = {\ vec {v}} t + {\ vec {r}} _ {0}}

.

Evenly accelerated movement :

{\ displaystyle {\ vec {r}} (t) = {\ frac {1} {2}} {\ vec {a}} t ^ {2} + {\ vec {v}} _ {0} t + { \ vec {r}} _ {0}}

.

If the (constant) acceleration and the initial speed are parallel or antiparallel, it is a uniformly accelerated or decelerated linear movement. Otherwise it is a parabolic movement such as a crooked throw .

{\ displaystyle {\ vec {a}}}

{\ displaystyle {\ vec {v}} _ {0}}

Harmonic oscillation , such as that carried out by the mass on a spring pendulum along the axis of the spring when it swings out of equilibrium : ${\ displaystyle \ vert {\ vec {A}} _ {0} \ vert}$ ${\ displaystyle {\ vec {r}} _ {0}}$

{\ displaystyle {\ vec {r}} (t) = {\ vec {r}} _ {0} + {\ vec {A}} _ {0} \ cdot \ sin (\ omega t)}

.

Individual evidence

↑ Rainer Müller: Classic mechanics: From long jump to Mars flight . Walter de Gruyter, 22 September 2010, ISBN 978-3-11-025003-9 , p. 58–.
^ Herbert A. Stuart, Gerhard Klages: Short textbook of physics . Springer-Verlag, March 14, 2013, ISBN 978-3-662-08228-7 , pp. 10–.
↑ Ekbert Hering, Rolf Martin, Martin Stohrer: Physics for engineers . Springer-Verlag, July 1, 2013, ISBN 978-3-662-09314-6 , p. 349–.

[Müller2010-1] Rainer Müller: Classic mechanics: From long jump to Mars flight . Walter de Gruyter, 22 September 2010, ISBN 978-3-11-025003-9 , p. 58–.

[StuartKlages2013-2] Herbert A. Stuart, Gerhard Klages: Short textbook of physics . Springer-Verlag, March 14, 2013, ISBN 978-3-662-08228-7 , pp. 10–.

[HeringMartin2013-3] Ekbert Hering, Rolf Martin, Martin Stohrer: Physics for engineers . Springer-Verlag, July 1, 2013, ISBN 978-3-662-09314-6 , p. 349–.