Way (physics)

A path of an object assumed to be point-like is the course of its location with advancing time as a result of its movement . The path is also known as a train ; it runs along a trajectory . The position on the way is described by a position vector relative to a freely selectable reference point, which is assumed to be stationary. The preferred symbol for the way is (from Latin spatium space, expansion, distance). ${\displaystyle s}$

The term "path" sometimes means its length along the trajectory. To distinguish this scalar variable is also referred to as the distance covered , the distance or the arc length .

Path as the course of the place

→ Main article : Trajectory (physics)

The path as the course of the location of a point-shaped object can be determined by calculations as the solution of an equation of motion , which can only be specified in a closed form in very simple cases, or by measurements e.g. B. determined by particles in a wire chamber . Either the time or the distance can be selected as parameters for the course: Optionally or . ${\displaystyle {\vec {r}}(t)}$ ${\displaystyle {\vec {r}}(s)}$

Path length

Position vector and path element when moving on a path

The path length (especially when waves or currents are considered, also called the run length ) from A to B is the sum of all distances between A and B. For sufficiently small, geometrically simple or straight path segments : ${\displaystyle \Delta s}$

${\displaystyle s_{\mathrm {AB} }=\sum _{A}^{B}\Delta s\qquad }$

When a physical body moves, its location changes continuously over time. The curve that he describes is called a trajectory . The scalar path element is the amount of the infinitesimal change in location : ${\displaystyle \mathrm {d} s}$ ${\displaystyle \mathrm {d} {\vec {r}}}$

${\displaystyle \mathrm {d} s=|\mathrm {d} {\vec {r}}|=|{\vec {v}}(t)|\;\mathrm {d} t}$

When calculating the path length, the summation then changes to an integration . This gives the length of the part of the trajectory covered between the two times:

${\displaystyle s=\int \limits _{(1)}^{(2)}\mathrm {d} s=\int \limits _{t_{1}}^{t_{2}}|{\vec {v}}(t)|\;\mathrm {d} t}$

In general, the path length is longer than the distance between the beginning and the end of the trajectory. ${\displaystyle s}$

A simplification results in a one-dimensional process: The vectors can be replaced by scalars . For example, in the case of a vertical throw upwards, the acceleration due to gravity , the initial speed at the initial point in time , the location-time function and the initial altitude apply ${\displaystyle g}$ ${\displaystyle v_{0}>0}$ ${\displaystyle t_{0}=0}$ ${\displaystyle h(t)}$ ${\displaystyle h=0}$

${\displaystyle v(t)=v_{0}-g\;t}$.

${\displaystyle h(t)=v_{0}\;t-{\frac {g}{2}}\;t^{2}}$.

The throw reaches a maximum height of rise ; there is . At this point its sign is reversed . The distance to the starting point is calculated as follows: ${\displaystyle h_{\mathrm {max} }={v_{0}}^{2}/(2g)}$ ${\displaystyle v=0}$ ${\displaystyle v}$

${\displaystyle s=\int \limits _{0}^{2v_{0}/g}|v(t)|\;\mathrm {d} t=2\int \limits _{0}^{v_{0}/g}(v_{0}-g\;t)\;\mathrm {d} t={v_{0}}^{2}/g}$

Path in a physical field

If an object is displaced in a physical field along a path from location A to location B, which is given by the location vectors and and a field force acts on the object , then work is done by the field ${\displaystyle {\vec {r}}_{A}}$ ${\displaystyle {\vec {r}}_{B}}$ ${\displaystyle {\vec {F}}}$ ${\displaystyle W_{\mathrm {AB} }}$

${\displaystyle W_{\mathrm {AB} }=\int _{\mathrm {A} }^{\mathrm {B} }{\vec {F}}\cdot \mathrm {d} {\vec {s}}}$.

If the field is homogeneous, it is a location-independent constant. Then applies ${\displaystyle {\vec {F}}}$

${\displaystyle W_{\mathrm {AB} }={\vec {F}}\cdot \left(\int _{\mathrm {A} }^{\mathrm {B} }\mathrm {d} {\vec {s}}\right)={\vec {F}}\cdot {\vec {r}}_{\mathrm {AB} }}$

since the integration of the vectorial path element results in the displacement vector from A to B. ${\displaystyle \mathrm {d} {\vec {s}}}$ ${\displaystyle {\vec {r}}_{\mathrm {AB} }}$

For example, a constant, homogeneous electric field with the field strength on a charge , dis moves in this field from to , does the work ${\displaystyle {\vec {E}}}$ ${\displaystyle q}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle W_{\mathrm {AB} }=q\int _{\mathrm {A} }^{\mathrm {B} }{\vec {E}}\cdot \mathrm {d} {\vec {s}}=q{\vec {E}}\cdot {\vec {r}}_{AB}}$

performed.

If a field is a source or potential field , then the force caused by it is a conservative force . The work involved in moving the body from one place to another then only depends on the position of the two places, but not on the path between them. This is what you mean when you talk about travel- independent work.

The same applies to a moving mass in a gravitational field . With the force acting on a time-independent mass , which is equal to mass times acceleration, so with results ${\displaystyle m}$ ${\displaystyle {\vec {F}}=m{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}}$

${\displaystyle W_{\mathrm {AB} }=m\int _{\mathrm {A} }^{\mathrm {B} }{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}\cdot \mathrm {d} {\vec {s}}=m\int _{\mathrm {A} }^{\mathrm {B} }{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}\cdot {\vec {v}}\,\mathrm {d} t=m\int _{\mathrm {A} }^{\mathrm {B} }{\frac {1}{2}}\,\mathrm {d} (v^{2})={\frac {m}{2}}{v_{\mathrm {B} }}^{2}-{\frac {m}{2}}{v_{\mathrm {A} }}^{2}}$.

The work depends only on the kinetic energy at the start and end point and not on the kinetic energy during the path.

Web links

Wiktionary: way  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ^ Walter Weizel: Textbook of theoretical physics: Volume 1 Physics of processes. Springer, 2nd edition 1955, p. 5
2. ^ Ernst Grimsehl, Kurt Altenburg: Grimsehl textbook of physics: Volume 1 Mechanics • Acoustics • Heat theory. Springer, 27th edition 1991, p. 27
3. ↑ Bruno Assmann, Peter Selke: Technical Mechanics 3: Volume 3: Kinematics and Kinetics. Oldenbourg, 14th edition 2007, p. 62
4. ↑ Gottfried Falk, Wolfgang Ruppel: Mechanics, Relativity, Gravitation: The Physics of the Scientist. Springer, 3rd ed. 1983, p. 23.
5. ^ Paul Dobrinski, Gunter Krakau, Anselm Vogel: Physics for engineers. Vieweg + Teubner, 12th edition 2010, p. 17.
6. ↑ Lothar Papula: Mathematics for Engineers and Natural Scientists Volume 3: Vector Analysis… Springer Vieweg, 7th edition 2016, p. 12 ff
7. ↑ Klaus Lüders, Robert O. Pohl (ed.): Pohl's introduction to physics: mechanics, acoustics and thermodynamics. Springer, 19th ed., P. 11.
8. ^ Helmut Lindner: Physics for Engineers. Vieweg, 12th ed. 1991, p. 34
9. ^ Walter Weizel: p. 10