# Calculating Tennis Ball Trajectory: Force and Integration Explained

• diegorodriguez
However, we can use numerical methods to approximate the positions of the tennis ball at each time t, given the total force exerted. This force is determined by considering the drag force, lift force, and gravitation force, which are all dependent on the speed, angular speed, radius, density, and mass of the ball. These forces can be calculated using the equations described in the provided resource. To determine the positions of the ball, we can use numerical integration techniques. This involves breaking down the motion into small time intervals and calculating the position at each interval based on the forces acting on the ball. While there is no exact solution, using numerical methods can provide a close approximation of the positions of the ball at each time t. In summary,

#### diegorodriguez

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?

As far as I am aware, there is no analytic solution for 2D motion with drag.

## 1. What factors affect the trajectory of a tennis ball?

The trajectory of a tennis ball is affected by various factors such as the angle of the racket at impact, the speed of the ball, air resistance, and the spin of the ball. Other factors such as wind, temperature, and altitude can also have an impact on the trajectory.

## 2. How does spin affect the trajectory of a tennis ball?

Spin plays a crucial role in the trajectory of a tennis ball. A topspin shot will cause the ball to dip more quickly, while a backspin shot will cause the ball to bounce higher. Sidespin can also cause the ball to curve in a certain direction, making it difficult for the opponent to return.

## 3. What is the optimal trajectory for a tennis serve?

The optimal trajectory for a tennis serve is a flat, fast trajectory with a slight topspin. This allows the ball to travel quickly and dip towards the end of its trajectory, making it difficult for the opponent to return.

## 4. How does air resistance affect the trajectory of a tennis ball?

Air resistance, also known as drag, can slow down the speed of a tennis ball and cause it to drop faster. This can result in a shorter trajectory than expected, especially on high-speed shots. Players must take air resistance into account when aiming for a specific trajectory.

## 5. Can the trajectory of a tennis ball be predicted accurately?

While there are many factors that can affect the trajectory of a tennis ball, it is possible to predict its trajectory with a high degree of accuracy using mathematical models and computer simulations. However, unpredictable factors such as wind and variations in court surfaces can make it challenging to predict the exact trajectory of a ball in real-time.