An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method
An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method
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DOI：
10.1088/1751-8113/40/29/015
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发表时间：
2007-07
期刊：
影响因子：
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通讯作者：
K. Yabushita;Mariko Yamashita;K. Tsuboi
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文献类型：
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作者：
K. Yabushita;Mariko Yamashita;K. Tsuboi
我们考虑二维抛射运动问题，其中作用于空中运动物体的阻力与物体速度的平方成正比(二次阻力定律)。众所周知，二次阻力定律在雷诺数为1×10~3∼2×10~5(例如球体)的范围内适用于实际情况，如投球。虽然已经用摄动法得到了小弹角情况下的一些解，但人们认为这种情况下的运动方程对于一般的弹丸角是不可解的。为了得到一般的解析解，我们将廖氏的同伦分析方法应用于该问题。与摄动法不同的是，同伦分析方法适用于不含小参数的问题。我们不仅将同伦分析方法应用于控制微分方程，而且还应用于速度向量的代数方程，以扩大收敛半径。最后，我们得到了该问题的解析解，并研究了该解的有效性。
We consider the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object (quadratic resistance law). It is well known that the quadratic resistance law is valid in the range of the Reynolds number: 1 × 103 ∼ 2 × 105 (for instance, a sphere) for practical situations, such as throwing a ball. It has been considered that the equations of motion of this case are unsolvable for a general projectile angle, although some solutions have been obtained for a small projectile angle using perturbation techniques. To obtain a general analytic solution, we apply Liao's homotopy analysis method to this problem. The homotopy analysis method, which is different from a perturbation technique, can be applied to a problem which does not include small parameters. We apply the homotopy analysis method for not only governing differential equations, but also an algebraic equation of a velocity vector to extend the radius of convergence. Ultimately, we obtain the analytic solution to this problem and investigate the validation of the solution.