Evenly accelerated movement

Evenly accelerated movement with an initial speed and initial path of zero: Path, speed and acceleration are plotted as functions of time.

A ball evenly accelerated downwards by the acceleration of gravity

A uniformly accelerated movement is a movement in which the acceleration is constant in terms of strength and direction. The smoothly accelerated motion is a straight line motion when the acceleration and initial velocity are collinear . If this is not the case, a parabola is created as a trajectory . By choosing an inertial system in which the initial speed is zero, you always get a straight line movement. When the acceleration becomes zero, the uniform motion is obtained .

Examples of an evenly accelerated movement are free fall or oblique throw without taking air resistance into account.

Laws

Reading of the acceleration a for a uniformly accelerated movement in the slope triangle .

If the uniformly accelerated movement is straight, you can use numbers (scalars) instead of vectors (scalar form) for calculations. It is sufficient to express the orientation of the speed and acceleration vector by the sign . One direction (usually the direction of movement) is marked as positive, the opposite direction as negative.

If the uniformly accelerated movement is not straight, the more general vector form must be used. The following laws apply:

Evenly accelerated movement
		Scalar form		Vector shape
necessary condition		${\displaystyle a={\text{const.}}}$		${\displaystyle {\vec {a}}={\text{const.}}}$
Speed-time law		${\displaystyle v(t)={\dot {s}}(t)=at+v_{0}}$		${\displaystyle {\vec {v}}(t)={\dot {\vec {s}}}(t)={\vec {a}}t+{\vec {v}}_{0}}$
Way-time law		${\displaystyle s(t)={\frac {a}{2}}t^{2}+v_{0}t+s_{0}}$		${\displaystyle {\vec {s}}(t)={\frac {\vec {a}}{2}}t^{2}+{\vec {v}}_{0}t+{\vec {s}}_{0}}$
used formula symbols
${\displaystyle a,{\vec {a}}}$	acceleration		${\displaystyle \left[{\frac {\text{m}}{{\text{s}}^{2}}}\right]}$	${\displaystyle {\begin{aligned}{\vec {a}}&={\frac {\mathrm {d} {\vec {v}}(t)}{\mathrm {d} t}}={\dot {\vec {v}}}(t)\\&={\frac {\mathrm {d} ^{2}{\vec {s}}(t)}{\mathrm {d} t^{2}}}={\ddot {\vec {s}}}(t)\end{aligned}}}$
${\displaystyle s(t),{\vec {s}}(t)}$	Position at the time ${\displaystyle t}$		${\displaystyle \left[{\text{m}}\right]}$
${\displaystyle s_{0},{\vec {s}}_{0}}$	Starting position (starting path) at the point in time ${\displaystyle t=0}$		${\displaystyle \left[{\text{m}}\right]}$	${\displaystyle {\vec {s}}_{0}={\vec {s}}(t=0)}$
${\displaystyle t}$	time		${\displaystyle \left[{\text{s}}\right]}$
${\displaystyle v(t),{\vec {v}}(t)}$	Speed ​​at the time ${\displaystyle t}$		${\displaystyle \left[{\frac {\text{m}}{\text{s}}}\right]}$	${\displaystyle {\vec {v}}(t)={\frac {\mathrm {d} {\vec {s}}(t)}{\mathrm {d} t}}={\dot {\vec {s}}}(t)}$
${\displaystyle v_{0},{\vec {v}}_{0}}$	Initial speed at the point in time ${\displaystyle t=0}$		${\displaystyle \left[{\frac {\text{m}}{\text{s}}}\right]}$	${\displaystyle {\vec {v}}_{0}={\vec {v}}(t=0)}$

Derivation

Out ${\displaystyle {\dot {\vec {v}}}={\vec {a}}}$

with constant acceleration, integration results in a speed that is linearly dependent on time:

${\displaystyle {\vec {v}}={\vec {a}}t+{\vec {v}}_{0}}$,

where is the constant of integration which includes the initial speed. ${\displaystyle {\vec {v}}_{0}}$

Since the velocity is the first derivative of the position with respect to time:

${\displaystyle {\dot {\vec {s}}}={\vec {v}}={\vec {a}}t+{\vec {v}}_{0}}$;

Subsequent integration results in the path-time law :

${\displaystyle {\vec {s}}(t)={\frac {\vec {a}}{2}}t^{2}+{\vec {v}}_{0}t+{\vec {s}}_{0}{\text{,}}}$

where is the starting position. ${\displaystyle {\vec {s}}_{0}}$

The equations for speed and position are thus:

${\displaystyle {\begin{aligned}&{\vec {v}}(t)={\vec {a}}t+{\vec {v}}_{0}\\&{\vec {s}}(t)={\frac {\vec {a}}{2}}t^{2}+{\vec {v}}_{0}t+{\vec {s}}_{0}\end{aligned}}}$

Web links

Wikibooks: Collection of formulas for classical mechanics  - learning and teaching materials

Individual evidence

1. ↑ Wolfgang Demtröder : Experimentalphysik 1: Mechanics and heat . limited preview in Google Book search.