Optimizing Serve Angle to Clear the Net in Tennis: A Practical Example

In this article, we will explore the physics behind serving a tennis ball with a precise angle to just clear the net. Given the initial speed, height of the serve, and distance to the net, we will use the principles of projectile motion to find the angle that ensures the ball clears the net. This example will help tennis players improve their serves and better understand the mechanics behind this fundamental skill.

Initial Conditions and Equation Setup

The given parameters for our problem are:

Initial speed of the ball: ( v_0 47.5 , text{m/s} ) Height of the serve: ( h_0 2.3 , text{m} ) Height of the net: ( h_{text{net}} 0.89 , text{m} ) Horizontal distance to the net: ( d 11.8 , text{m} )

We need to find the angle ( theta ) such that the ball just clears the net. We will use the equations of projectile motion to solve this problem.

Vertical and Horizontal Motion Equations

Vertical Motion Equation

The vertical position ( y_t ) of the ball as a function of time ( t ) is given by:

[ y_t h_0 v_{0y} t - frac{1}{2} g t^2 ]

where ( g 9.81 , text{m/s}^2 ) is the acceleration due to gravity and ( v_{0y} v_0 sin theta ) is the initial vertical component of the velocity.

Horizontal Motion Equation

The horizontal position ( x_t ) of the ball is given by:

[ x_t v_{} t ]

where ( v_{} v_0 cos theta ).

Finding the Time to Reach the Net

The time ( t ) to reach the net can be found from the horizontal motion equation:

[ t frac{d}{v_{}} frac{11.8}{v_0 cos theta} frac{11.8}{47.5 cos theta} ]

Substituting ( t ) into the vertical motion equation gives us:

[ y_t h_0 v_{0y} t - frac{1}{2} g t^2 ]

Substituting ( v_{0y} ) and ( t ) into the equation:

[ h_{text{net}} h_0 v_0 sin theta left( frac{11.8}{47.5 cos theta} right) - frac{1}{2} g left( frac{11.8}{47.5 cos theta} right)^2 ]

Substituting the known values:

[ 0.89 2.3 times 47.5 sin theta left( frac{11.8}{47.5 cos theta} right) - frac{1}{2} times 9.81 times left( frac{11.8}{47.5 cos theta} right)^2 ]

After simplifying the equation:

[ 0.89 2.3 times 11.8 tan theta - frac{9.81 times 11.8^2}{2 times 47.5^2 cos^2 theta} ]

Rearranging the equation:

[ 11.8 tan theta - frac{9.81 times 11.8^2}{2 times 47.5^2 cos^2 theta} 0.89 - 2.3 ]

Solving for ( theta ) is complex and typically requires numerical methods or graphing techniques to find the precise value.

Approximate Solution

Using numerical or graphical methods or graphing tools, we can find the angle ( theta ) that satisfies the equation. A numerical solution gives:

[ theta approx 8.5^circ ]

Thus, the angle at which the ball must be served to just cross the net is approximately ( 8.5 ) degrees below the horizontal.

Conclusion

Understanding the principles of projectile motion helps tennis players optimize their serving angle to maximize the chance of clearing the net. This practical example demonstrates how to calculate the necessary angle using basic physics, providing valuable insight for players at all levels.