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Ecuaciones de cinemática por integración - Problemas y ejercicios de matematica

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Deducción de las ecuaciones

de la cinemática por Integración

 

 

Nuestro trabajo ha de comenzar con la suposición de que ya existe un estudio básico de las ecuaciones del movimiento, o al menos una noción, vale de igual manera detallar lo siguiente: Sabemos que o sea que la aceleración es la derivada de la velocidad con respecto al tiempo, vamos entonces a proceder con el desarrollo:

Multiplicamos ambos miembros de la igualdad por dt obteniendo integramos ambos miembros para obtener: al operar obtenemos ahora podemos integrar: o sea

Muy bien nuestro trabajo está a la mano, ahora también podemos afirmar que: siguiendo entonces el mismo proceder: integrando

de allí usando nuestro resultado anterior ordenamos __ya casi está lista o como la conocemos mejor:

 

Si el movimiento fuera vertical, sólo debemos cambiar la x por la y en las ecuaciones y tenemos las ecuaciones para movimientos en el eje y con aceleración conocida (g=-9.8 m/s²).

¿La altura máxima?

Muy bien sabemos que el máximo se obtiene en el cero de la derivada de una función: ok entonces nos dará el tiempo en el cual la altura es máxima igualamos y a 0 para obtener el máximo y nos da que sustituímos en la ecuación original el tiempo obtenido y tenemos y así   con lo cual obtenemos

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