Table of Contents

## Rationale:

This investigation will consider the dynamics of a volleyball after it has been served. The reason I chose this topic is because I love to play volleyball, and I do it almost everyday. A while ago I suffered from a back injury that prevented me from serving. Now that it’s mostly healed, I’m beginning to serve again. Now that serving is an option for me, if I want to actually serve in a game, I have to prove to my coach that I can serve effectively. Some factors that greatly impact the effectiveness of a serve include how high the ball goes, the time period the receiving team has to move to the ball, and the velocity of the ball.

## Aim

My aim is to determine the difference in the dynamics of a volleyball when it is served using various methods. Methods of serving include the underhand serve, standing topspin, standing float, jump topspin, and jump float. By filming my teammates’ serves and analyzing the video footage using Logger Pro, I was able to create graphs that model the path of the ball for each type of serve.[*] The following image depicts how I evaluated the serve by placing a point where the ball is located in each frame of the video to make Position vs. Time graph for an underhand serve. The picture is just a snapshot of process, so it’s not a complete graph yet.

## Underhand Serve: Sample Calculation

An underhand serve is where the server holds the ball in one hand at about waist-height and hits the bottom of the ball with the other fist. This is a very basic method of serving, but it will act as a good basis of comparison for the other types of serves.

Equation that models the Position vs. Time for underhand: Maximum height. This impacts the difficulty of the serve, because the higher the ball goes, the easier it is for the defender to get to it.

Alternatively, this value could be found by finding the value for which the derivative of this function is equal to zero, because the ball is instantaneously at rest when it reaches it’s maximum height.

## Velocity

The velocity of the volleyball is an important aspect of how difficult a serve is to receive; the faster the ball is going, the harder it is for a defensive player to make an accurate pass. The velocity also impacts the amount of time the defensive player has to move to the ball, which will be evaluated later. In order to determine the vertical component of the velocity of the ball, the derivative must be taken of the Position vs. Time graph (this graph only accounts for vertical position, thus taking the derivative will only yield the vertical component of velocity).

This component of the velocity of the ball when it is at a height of 3 feet will be evaluated. This height was chosen because this is the approximate height at which the defense on the opposing team will come in contact with the ball.

Solve for using the quadratic formula: The value for which the serve-receiver will come in contact with the ball cannot be negative, therefore the velocity of the ball when the receiver passes it will occur at seconds.

Thus of the volleyball when the opposing team receives it is. The horizontal component of the velocity remains unknown, but it can be calculated using trigonometry.

The value for ? is unknown, and can be calculated using the following formula where g is the acceleration due to gravity, v is the resultant velocity at , and is the time for which the volleyball is airborne (the time it takes for the volleyball to hit the ground). The following equation gives the value for at the launch point, so the value can be directly substituted into the equation this will work, because the horizontal velocity does not change in this model.

Time

Another important factor in a serve’s effectiveness is how long of a time the defensive team has to reach and pass the ball after it has been served. There are two ways to measure this. The first is by simply calculating how long the ball is in the air. This can be determined by finding the time at which the opposing team receives the ball. This has already been determined to be 1.82336 seconds. This assumes that the receiving team has full vision of the ball from the moment it is served, and can immediately tell where the ball is going and move to that location. However, due to opposing players blocking the defense’s vision, and possible confusion as to where the ball will actually land, especially if wind is a factor, it is extremely difficult for a defensive player to know where to go immediately when the ball is served. Thus, a more accurate way to determine the time a defensive player can use to move to the ball is to find the period of time for which the ball is in the air after it has passed over the net. This works because, even if the defensive player can see the ball before it comes over the net, they usually take this time to process where it is going, and they start to move after the ball goes over the net. This does not necessarily occur when the ball is at it’s peak height, thus the time at which the ball has a horizontal displacement of 30 feet must be calculated (this is the distance between the end of the court, where the volleyball is served, and the net).

Displacement formula: We assume no horizontal acceleration of the ball, so the equation becomes. The time the receiver has to move to the ball is the difference in time between when the ball passes over the net and when it is received:

This time doesn’t seem quite right, considering the fact that from the picture it looks as though the ball crosses the net at what looks to be very close to the maximum height, which occurs at . What this tells me is that the horizontal velocity must be incorrect. I’ll continue to use these mathematical processes with the other types of serves and record the calculations, and hopefully the results will help me determine what exactly it is that caused the horizontal velocity to be incorrect.

## Graphs of other Serves

Standing Topspin. Standing topspin is an overhand serve where the server contacts the ball with their hand approximately one arm’s length above their head. The server snaps her wrist when coming in contact with the ball, which gives the ball a forward spin, as indicated by the name.

## Standing Float

A standing float serve is an overhand serve that is very similar to a standing topspin serve. The only difference is that instead of snapping their wrist, the server hits the center of the ball with a firm hand, which results in the ball having little to no spin as it goes over the net. Because it has no spin, air currents will often cause the ball not to move in a directly straight path, making it look like it’s “floating around.”

## Jump Topspin

In a jump topspin serve, the server throws the ball above their hand and takes a four-step approach and jumps to hit the ball, snapping their wrist as they come in contact with it. This motion is very similar to spiking the ball, which allows the player to hit the ball with extreme topspin—much more so than from a standing position.

## Equation that models Position vs. Time for jump topspin:

It’s interesting that the leading coefficient is so much higher than any of the other graphs. This is likely due to the forward spinning motion the ball has, which causes the ball to land much closer from where it was served than it would otherwise. So it makes sense that the leading coefficient would be larger, because it vertically stretches the graph.

Jump Float. For a jump float serve, the server generally takes fewer steps in an approach than in a jump topspin serve, and they hit the ball with a firm hand so the ball has no spin.

## Equation that models Position vs. Time for jump float:

The constant for this graph is lower than for jump topspin, which is likely because a jump float requires fewer steps leading up to when the server jumps, resulting in a smaller jump. However, the constant for both jump topspin and jump float are higher than all of the other graphs because the ball was hit from a higher point.

## Evaluating and Comparing the Serves

Now that I have my method of calculating all of the necessary values and a model for each type of serve, I can compare the data. Based on the values for the time the defensive player has to move to pass the ball, it seems the calculations must be incorrect; there’s no way that the player can have a negative amount of time. This value was largely dependent upon the changing horizontal component of the velocity, which in ideal circumstances would be constant, but for this investigation, it seems that it was in fact necessary to take into account. When using the equation I used a value of zero for the horizontal acceleration, but this clearly was a source of error in the calculations. Thus, to achieve accurate results, I would have to create a model that accounts for air resistance to accurately determine both the horizontal velocity of the volleyball and how much time a defensive player has to move to pass the volleyball.

After all of that work, it was pretty disappointing to see that the only thing I would be able to evaluate regarding the serves would be the maximum height of the serve and the approximate vertical velocity. However, I later realized that I had overlooked something in my graphs. Logger Pro has a capability that can allow me to determine the actual horizontal and vertical velocities of the volleyball, so I can at least use the data from this to compare the types of serves.

## Conclusion: Deciding the Best Method of Serving

Jump topspin has a significantly higher resultant velocity than all of the other types of serves, the rest of them are about the same except for the underhand serve. Another connection that can be drawn is that jump serves have higher velocities than standing serves. Additionally, topspin serves are faster than their float serve counterparts.

The time the ball is in the air for correlates with the velocity of the ball—topspin jump has the best results with the shortest time, then jump float, standing topspin, float, and underhand respectively. The maximum height of each serve is about the same, except for underhand, which is much higher and jump float, which is slightly lower than the rest. The lower the maximum height of the serve the better, because the ball is easier to move to if it goes higher, so the jump float is most effective in this regard.

Based on these results, I think that most effective serve is likely the jump topspin, because its velocity is so much greater and the time the ball is in the air is so much lower than the other types of serves. Jump topspin is one of the more difficult serves to master, so I had always been apprehensive to undertake learning it. But now I’m glad I know how much more effective it will likely be based on my results from this investigation, so I know it will be worth it!