miércoles, 9 de mayo de 2007

¿Cuál es la distancia al horizonte?

Imagina que estás en la playa de pie mirando el horizonte: ¿a qué distancia se encuentra? Antes de seguir haz una estimación: ¿5 Km?, ¿50?, ¿500?...


El horizonte es la línea a partir de la cual no podemos ver más allá a causa de la curvatura de la Tierra. Entonces la línea visual que une nuestros ojos con el horizonte es una línea recta tangente a la Tierra, y por tanto perpendicular al radio de esta en el horizonte.

Los datos necesarios son el radio de la Tierra(r=6378 Km. aproximadamente en el ecuador), y la altura a la que se encuentran los ojos del observador ("a" en el esquema, claramente exagerado). Aplicando el teorema de Pitágoras calculamos la distancia al horizonte (h):
Si suponemos que nuestros ojos se elevan a=1,70 metros del suelo (0,0017 Km.) y aplicamos la fórmula el horizonte estará a 4,66 Km. Dependiendo de nuestra altura, este valor puede oscilar entre 4 Km(para los más pequeños de la casa) y 5 (para los jugadores de baloncesto).

Y yo me pregunto... ¿a qué distancia está 'vuestro horizonte'?

221 comentarios:

1 – 200 de 221   Más reciente›   El más reciente»
koinor dijo...

Y pensar que en el oriente se pensaba que en el horizonte te caias a un precipicio. Honestamente siempre pensé que eran 10km lo que se veia. ¿realmente es tanta la diferencia entre un niño y un jugador de baloncesto?

Lobo dijo...

Hola, la verdad es que nunca me había planteado cuál podría ser la distancia al horizonte...

Por cierto, ¿hay alguna manera de poder contactar contigo, aparte de en los comentarios de este blog? Gracias

Sable dijo...

Así es, la diferencia es bastante considerable dependiendo de la altura.

Se puede hacer una aproximación realmente buena, omitiendo el a^2, que es un número insignificante(+ 0,000002...). Operando (sqrt2=1,41...) y (sqrt(r)=79,86...), quedaría:

h=112,94*(sqrt(a))

Recordar que la 'a' es la altura a la que se encuentran nuestros ojos y no nuestra altura dicha propiamente. Además esta altura se ha de pasar a Km. dividiendo por 1000.

Sable dijo...

Hola Lobo.

Curioso ver como todo resulta lógico. Ya sabemos que si vemos un barco en el horizonte en la playa, por muy altas que sean las velas y lo seamos nosotros, no podrá estar a más de 10 Km.

Por supuesto. Sería interesante conversar contigo. Agregame si quieres a ari_finlandia@hotmail.com

Sable dijo...

A mi también me sorprendió que la lejanía al horizonte pueda variar tanto entre personas. Para convencer de la diferencia que hay entre bajitos y altos:

Supogamos que tenemos una persona con los ojos a una altura de 1m(a) del suelo y otra a 2m(2a), el doble.

para el más bajo
h=112,94*(sqrt(a))

y para el más alto
h=112,94*(sqrt(2a))
sacamos el 2 de la raíz cuadrada
h=112,94*1,41*(sqrt(a))

En esto último se ve que la distancia se multiplica por 1,41, es decir, aumenta en un 41%.

Para 1m(o,oo1 Km.) sería 3,5Km.
Para 2m(0,002 Km.) sería 5Km., un 41% más aproximado.

Carlos dijo...

La cuestión del horizonte es quizás más interesante si se plantea en sentido inverso. ¿Cuál debe ser la altura de un objeto para que lo veamos a X kilómetros? El análisis es el mismo (despreciando la altura del observador, claro), ya que lo único que varía es que estamos situados en el otro vértice del triángulo.

Sable dijo...

Hola Carlos.
Pues si que resulta más interesante lo que dices. Aunque al despreciar nuestra altura podría dar lugar a errores de aproximación, pero quedaría más o menos reflejado.

¿Cuál debe ser la altura de un objeto para que lo veamos a X kilómetros(sin despreciar la altura del observador)?

Lo he estado pensando y trabajando en ello, y la verdad que se complica. Tras un par de horitas de trabajo, mis resultados:

Sable dijo...

¿Cuál debe ser la altura de un objeto para que lo veamos a X kilómetros(sin despreciar la altura del observador)?

Sea X=distancia al objeto.
Sea Y=altura del objeto.
Sea a=altura de los ojos del observador

La idea es casi la misma que el problema del post, pero:

En el dibujo hay que prolongar 'h' hacia abajo, añadiéndole d. Cuánto valga d va a depender del valor de Y. Desde el nuevo extremo de h trazamos un segmento al centro terrestre de longitud r+Y. Tenemos entonces dos triángulos rectángulos independientes:

-En el primero, la incógnita es 'h' (va a depender de la altura del observador, 'a'). Esto es el mismo cálculo realizado en el post.

-En el segundo triángulo, 'a' se reemplaza por 'Y'. Siendo ahora 'Y' la incógnita. La distancia del objeto al horizonte común la sabemos es d=X-h.
Para saber 'd' se necesita saber 'h' y para ello resolver primero el otro triángulo.

Con esto último tenemos la altura del objeto, 'Y'.


He intentado simplificar la explicación al máximo. Se aconseja seguir la explicación con lápiz y papel.

Saludos!:)

G dijo...

lindo post! van tres detalles:

1) hay una forma sencilla de resolver el problema inverso: la recta que ve la persona "a" siempre será tangente al círculo en el mismo punto con el que calculás h, sólo que ahora se puede prolongar hacia abajo, y a medida que te alejes, va aumentando la altura mínima que puede ver del otro objeto. Por ejemplo, se podría mirar a los ojos con otra persona de su misma altura ubicada a 9,3 km.

2) ¿Cuál es la distancia 'real' sobre la Tierra? [se puede decidir rápidamente si es más grande o más chica recordando la demostración del límite sen(x)/x -> 1 si x -> 0]

3) está muy bien plantearlo en el mar! en tierra firme no vale!

Sable dijo...

Gracias por tu atención G.
1)Lo que dices es cierto y corrige aquello que dige en el 4º comentario "Ya sabemos que si vemos un barco en el horizonte en la playa, por muy altas que sean las velas y lo seamos nosotros, no podrá estar a más de 10 Km" y con tus amables cálculos sabemos que no es verdad. Los 9,3 km corresponderían a una persona que tiene los ojos a 1.7m del suelo.

2)Muy buena observación. Aunque en este caso puede parecer complicado, aplicando lo que tu dices (comparando 2tgx y 2arcotgx) resulta fácilmente visible. Aunque no he llegado aún en la escuela a este apartado, me he adelantado.

3) Un pequeño matiz.:)

JuanJe dijo...

Pero además, tienes que tener en cuenta efectos de la optica, como son si es la refracción, dependiendo de como este la superficie (atmosfera) de la zona en la que estes.

No estoy diciendo que suceda siempre

Sable dijo...

Gracias por el matiz Juanje. Así es la aplicación de las matemáticas al medio :)

Néstor dijo...

Bien, claro... para mayor claridad de los novatos tendrías que incluir el desarrollo del binomio.
Les dejo la formula para planilla de calculo. Ponerla en cualquier celda menos en A1 donde se debe poner la altura del observador en metros.

=RAIZ(A1^2+(A1*12756000))

Atenciòn; Dado lo sencible del asunto altura del observador recomiendo no poner la altura del mismo sino, la distancia del suelo a los ojos.

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Anónimo dijo...

Estimados amigos, el cálculo de la distancia del horizonte tiene una impresición de relevancia de acuerdo las medidas geodésicas. La tierra no es una esfera perfecta, es màs bien un elipsoide, y siendo el radio polar más corto que el radio ecuatorial, la distancia del horizonte depende de la latitud a la que se encuentre el observador.
Dato a tener en cuenta. Podemos calcular un delta o variable y relacionarlo con la latitud considerando el radio más largo versus el más corto.
Para resumir, creo que el calculo debe hacerse en millas náuticas, siendo 1Mn = 1.852,216 Metros.
Una milla náutica equivale a la longitud de un arco de 1' (un minuto, la sesentava parte de un grado) de meridiano terrestre.

De todas maneras el cálculo descrito es bastante certero, la diferencia en MNs. no supera el 10%.
Saludos,
Mario

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