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Flight and bounce of spinning sports balls
American Journal of Physics ( IF 0.9 ) Pub Date : 2020-11-01 , DOI: 10.1119/10.0001659
Jacob Emil Mencke 1 , Mirko Salewski 1 , Ole L. Trinhammer 1 , Andreas T. Adler 2
Affiliation  

Standard university or high-school physics teaching material on projectile motion is usually based on Newton's second law in vacuum, neglecting aerodynamics. We present a low-cost experiment for teaching projectile motion using the students' cell phones and sports equipment, which allows the students to test theory and numerical simulation against experimental data in the real world. For a shot put, theoretical predictions assuming projectile motion in vacuum agree with experimentally obtained trajectories in air to within a few centimeters. However, for a table tennis ball, vacuum trajectories can be almost three times as long as experimentally obtained trajectories. An equation of motion including the aerodynamic drag force has no analytic solution, but it is straightforward to integrate numerically for high-school or first-year university students. Accounting for aerodynamic drag substantially improves the match with experimental data for any ball. In a second experiment, balls are shot with spin resulting in curveball trajectories. Numerical simulations including the Magnus force can give accurate predictions of 3D curveball trajectories, both curving according to the normal and the inverse Magnus effect. Balls shot with topspin and backspin are also accurately modelled. Finally, we model the bounce of an arbitrarily spinning ball using linear and angular impulse-momentum theorems and coefficients of restitution in vertical and horizontal directions. We find agreement with experimental data to within centimeters.

中文翻译:

旋转运动球的飞行和弹跳

关于弹丸运动的标准大学或高中物理教材通常基于真空中的牛顿第二定律,而忽略了空气动力学。我们提出了一个低成本的实验,使用学生的手机和运动器材来教授弹丸运动,让学生可以根据现实世界中的实验数据来检验理论和数值模拟。对于铅球,假设弹丸在真空中运动的理论预测与实验获得的空气中轨迹一致,误差在几厘米以内。然而,对于乒乓球,真空轨迹几乎是实验获得轨迹的三倍。包含气动阻力的运动方程没有解析解,但对高中或大学一年级学生进行数值积分很简单。考虑到空气动力阻力,大大提高了与任何球的实验数据的匹配。在第二个实验中,球被旋转射出,从而产生曲线球轨迹。包括马格努斯力在内的数值模拟可以准确预测 3D 曲线球轨迹,根据法向和逆马格努斯效应弯曲。上旋和后旋击出的球也被精确建模。最后,我们使用线性和角脉冲动量定理以及垂直和水平方向的恢复系数对任意旋转的球的弹跳进行建模。我们发现与实验数据的一致性在厘米以内。考虑到空气动力阻力,大大提高了与任何球的实验数据的匹配。在第二个实验中,球被旋转射出,从而产生曲线球轨迹。包括马格努斯力在内的数值模拟可以准确预测 3D 曲线球轨迹,根据法向和逆马格努斯效应弯曲。上旋和后旋击出的球也被精确建模。最后,我们使用线性和角脉冲动量定理以及垂直和水平方向的恢复系数对任意旋转的球的弹跳进行建模。我们发现与实验数据的一致性在厘米以内。考虑到空气动力阻力,大大提高了与任何球的实验数据的匹配。在第二个实验中,球被旋转射出,从而产生曲线球轨迹。包括马格努斯力在内的数值模拟可以准确预测 3D 曲线球轨迹,根据法向和逆马格努斯效应弯曲。上旋和后旋击出的球也被精确建模。最后,我们使用线性和角脉冲动量定理以及垂直和水平方向的恢复系数对任意旋转的球的弹跳进行建模。我们发现与实验数据的一致性在厘米以内。包括马格努斯力在内的数值模拟可以准确预测 3D 曲线球轨迹,根据法向和逆马格努斯效应弯曲。上旋和后旋击出的球也被精确建模。最后,我们使用线性和角脉冲动量定理以及垂直和水平方向的恢复系数对任意旋转的球的弹跳进行建模。我们发现与实验数据的一致性在厘米以内。包括马格努斯力在内的数值模拟可以准确预测 3D 曲线球轨迹,根据法向和逆马格努斯效应弯曲。上旋和后旋击出的球也被精确建模。最后,我们使用线性和角脉冲动量定理以及垂直和水平方向的恢复系数对任意旋转的球的弹跳进行建模。我们发现与实验数据的一致性在厘米以内。
更新日期:2020-11-01
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