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Analytical construction of the projectile motion trajectory in midair
Construcción analítica de la trayectoria del movimiento de un proyectil en el aire
Keywords:
projectile motion, construction of the trajectory, vertical asymptote (en)Movimiento de proyectiles, construcción analítica de la trayectoria, asíntota vertical (es)
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A classic problem of the motion of a projectile thrown at an angle to the horizon is studied. Air resistance force is taken into account with the use of the quadratic resistance law. An analytic approach is mainly applied for the investigation. Equations of the projectile motion are solved analytically for an arbitrarily large period of time. The constructed analytical solutions are universal, that is, they can be used for any initial conditions of throwing. As a limit case of motion, the vertical asymptote formula is obtained. The value of the vertical asymptote is calculated directly from the initial conditions of motion. There is no need to study the problem numerically. The found analytical solutions are highly accurate over a wide range of parameters. The motion of a baseball, a tennis ball, and a shuttlecock of badminton are presented as examples.
Se estudia un problema clásico del movimiento de un proyectil lanzado formando un ángulo con el horizonte. Se tiene en cuenta la fuerza de rozamiento del aire con el uso de la ley de resistencia cuadrática. En esta investigación se aplica principalmente una aproximación analítica. Las ecuaciones del movimiento del proyectil se resolvieron analíticamente para un largo periodo de tiempo arbitrario. Las soluciones analíticas construidas son universales, es decir, pueden ser utilizadas para condiciones iniciales de lanzamiento. Como un caso límite de movimiento, se obtiene la fórmula de asíntota vertical, cuyo valor se calcula directamente de las condiciones iniciales de movimiento. No hay necesidad de estudiar el problema numéricamente. Las soluciones analíticas encontradas son altamente precisas en un amplio rango de parámetros. El movimiento de una pelota de baseball, de tenis y un volante de bádminton se presentan como ejemplos.
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References
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