Horizontal throw

Horizontal litter (fountain in the garden of Belvedere Palace , Vienna, Austria).

In physics, the horizontal (also horizontal ) or horizontal throw is the process of movement that a body performs when it is thrown or shot parallel to the horizon, i.e. when it moves at a horizontal starting speed only under the influence of its weight . The resulting trajectory is a trajectory parabola with the point of throw as the vertex .

The horizontal throw can be broken down into two partial movements according to the principle of superposition ( principle of independence), the movement in the x- direction and in the y- direction do not influence each other. However, this only works if you consider the throw under idealized conditions, i.e. without taking air resistance into account .

Typical examples are the throwing of a body with a horizontal initial velocity in a gravity field (impact of a ball from a table, water jet splashing out of a horizontally held hose, dropping of a body from a horizontally flying airplane) or the trajectory of a charged particle in a homogeneous electrical field (e .g . Electron in the plate capacitor of a cathode ray tube ).

Analysis of the movement

In (horizontal) x-direction

For example, if a body is hit by a table, it leaves it with the constant velocity v ₀ in the horizontal direction. The laws of uniform motion apply to this component of motion :

${\displaystyle s_{\mathrm {x} }=x=v_{0}t\,}$ (Location coordinate),

${\displaystyle v_{\mathrm {x} }=v_{0}\,}$(Speed ​​in x direction), as well as

${\displaystyle a_{\mathrm {x} }=0\,}$(Acceleration in the x direction).

In the (vertical) y-direction

At the same time, the body falls down from the (initial / starting) height h ₀ . The laws of free fall apply , the body moves with constant acceleration (the acceleration due to gravity g):

${\displaystyle s_{\mathrm {y} }=y=-{\frac {gt^{2}}{2}}+h_{0}\,}$ (Location coordinates),

${\displaystyle v_{\mathrm {y} }=gt\,}$(Speed ​​in y direction) and

${\displaystyle a_{\mathrm {y} }=g\,}$(Acceleration in y -direction).

Equation of the trajectory parabola

Graph of the trajectory parabola

For the equation of the trajectory parabola (trajectory curve or locus curve), the orbit trajectory , the s _x equation is solved for t and the term for t is inserted into the s _y equation. How to get:

${\displaystyle s_{\mathrm {y} }=-{\frac {g}{2}}\left({\frac {s_{\mathrm {x} }}{v_{0}}}\right)^{2}+h_{0}=-{\frac {gs_{\mathrm {x} }^{2}}{2v_{0}^{2}}}+h_{0}\,}$,

or.

${\displaystyle y=-{\frac {g}{2}}\left({\frac {x}{v_{0}}}\right)^{2}+h_{0}=-{\frac {gx^{2}}{2v_{0}^{2}}}+h_{0}}$.

Generally one writes:

${\displaystyle s_{\mathrm {y} }=const\cdot s_{\mathrm {x} }^{2}+h_{\mathrm {0} }}$.

Throw range

With this you can insert the formula for the throwing time into the equation and get the throwing distance: ${\displaystyle s_{\mathrm {x} }}$

${\displaystyle s_{\mathrm {w} }=s_{\mathrm {x\,w} }=x_{\mathrm {w} }=s_{\mathrm {x\,max} }=x_{\mathrm {max} }=w=W=v_{0}{\sqrt {\frac {2h_{0}}{g}}}\,.}$

Throw height

The subtrahend (or summand) in the equation of the trajectory corresponds to the throwing height (starting height) if the throwing distance s _{w is} used for s _x :

${\displaystyle h_{0}=h_{\mathrm {max} }=s_{\mathrm {h} }=s_{\mathrm {y\,h} }=s_{\mathrm {y\,max} }=y_{\mathrm {max} }=H={\frac {gs_{\mathrm {w} }^{2}}{2v_{0}^{2}}}\,.}$

Throwing time

If you set the s _y equation and solve it for t , you get the time (throwing time, throwing time, flight time, flight time) of the body before it falls to the ground: ${\displaystyle 0\,}$

${\displaystyle t_{\mathrm {w} }=t_{\mathrm {f} }=t_{\mathrm {h} }=t_{\mathrm {max} }=T={\sqrt {\frac {2h_{0}}{g}}}\,.}$

Impact angle

If you denote the angle of the trajectory to the horizontal with , then you can calculate this angle from the following relationship: ${\displaystyle \beta \,}$

${\displaystyle \tan \beta ={\frac {v_{\mathrm {y} }}{v_{\mathrm {x} }}}\,.}$

Track speed

The amount of the path speed (and the impact or final speed at the end of the throwing time) can be calculated using the Pythagorean theorem : ${\displaystyle v(t)\,}$

${\displaystyle v(t)={\sqrt {v_{\mathrm {x} }^{2}+v_{\mathrm {y} }^{2}}}={\sqrt {v_{0}^{2}+(gt)^{2}}}\,}$

such as

${\displaystyle v_{\mathrm {max} }=v_{\mathrm {a} }=v_{\mathrm {e} }={\sqrt {v_{0}^{2}+(gt_{\mathrm {w} })^{2}}}\,}$.

Real case

See also : trajectory parabola with air resistance  and fall with air resistance

If friction has to be taken into account, an approximate calculation of the path can still be made at school level, for example using the small steps method .

Web links

• Experiments and exercises for the horizontal throw ( LEIFI )