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#ADVANCED PROJECTILE MOTION EQUATIONS FULL#
Examples are eventually provided and discussed for the accuracy and reliability of the given formulae against the full numerical simulation.ĭ. The presented formulae can be employed to estimate and optimize the characteristic kinematics of the projectile. Otherwise, full solution is further representable under the restriction of a uniform Magnus force with a still quadratic drag force. In the case of simultaneous existence of quadratic drag and Magnus forces together, the governing motion equation is highly nonlinear, permitting only to the analytical perturbation solutions. An expression for the vertical distance of the projectile thrown from a fixed position is also accounted for measuring the impacts of Magnus effects on the maximum height, the striking velocity to the ground and angle of stroke of the projectile during motion. Closed-form solutions for the speed of the object are provided either when the quadratic drag force is negligible or when the quadratic Magnus force is negligible. The angle of the trajectory in a given point is the same as the angle that the velocity vector form with the horizontal at that point.In this paper, it is targeted to present exact and approximate solutions to the motion of fired projectile in air subject to the Magnus effect. Once the strong> flight time is obtained, simply substitute in the equation of position of the horizontal component. It is the maximum horizontal distance, from the starting point of the motion to the point in which the body hits the ground. That is, the flight time is the time required for the height to become 0 (the projectile reaches the ground). It is calculated for y = 0, the vertical component of the position.
From that time, and from the equations of position, we can calculate the distance to the origin in the both axes, the x-axis and y-axis. Starting from the equation of velocity in the y-axis, and making v y = 0, we get the time t that it takes the body in get to this height. This value is reached when the velocity in the y-axis, v y , is 0. On the other hand, frequently in exercises, you would be asked for some of the following values. On the other hand, to know which trajectory the body follows, that is, its equation of trajectory, we can combine the above equations to eliminate t, getting:Īs expected, this is the equation of a parabola. The equation of position of the projectile motion is given by:
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