Calculating Tennis Ball Trajectory: Force and Integration Explained

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SUMMARY

This discussion focuses on calculating the trajectory of a tennis ball by analyzing the forces acting on it, specifically Drag force (D), Lift force (L), and Gravitational force (W). The equations provided for these forces are derived from physics principles, with D and L depending on variables such as speed (v), angular speed (w), radius (R), density (d), and mass (m). The integration of these forces over time is necessary to determine the x and y positions of the ball, particularly when considering the serve angle (Ro) and serve speed (vi). The participants note the absence of an analytic solution for 2D motion with drag, indicating the complexity of the problem.

PREREQUISITES
  • Understanding of classical mechanics, particularly forces acting on objects in motion.
  • Familiarity with calculus, specifically integration techniques.
  • Knowledge of fluid dynamics, especially drag and lift forces.
  • Basic physics concepts related to projectile motion.
NEXT STEPS
  • Study the integration of forces in motion, focusing on numerical methods for 2D trajectories.
  • Explore computational physics tools such as MATLAB or Python for simulating projectile motion with drag.
  • Research the effects of varying serve angles and speeds on tennis ball trajectories.
  • Investigate advanced topics in fluid dynamics related to sports physics, particularly in tennis.
USEFUL FOR

Physics students, sports scientists, engineers, and anyone interested in the mathematical modeling of projectile motion, particularly in the context of tennis ball dynamics.

diegorodriguez
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1. How to get x,y positions for a tennis ball at each time t given total force exerted?2. We consider Drag force D, Lift Force L and Gravitation force W

D=1/2*(0.55+1/((22.5+4.2*(v/(w*R))^2.5)^0.4))*pi*R^2*d*v^2
L=1/2*(1/(2+(v/(w*R))))*pi*R^2*d*v^2
W=m*g
​
as described in
http://www.physics.usyd.edu.au/~cross/TRAJECTORIES/42. Ball Trajectories.pdf

where v, speed; w, angular speed;R, radius;d, density;m, mass

3. I suppose we need to integrate force twice on time t for Fx and Fy given a Serve Angle Ro and Serve Speed vi, but how?
 
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As far as I am aware, there is no analytic solution for 2D motion with drag.
 

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