The physics behind a **tennis serve** is complex, and therefore challenging to master. The velocity, acceleration, and deceleration of the ball as it leaves the racket’s hitting surface is what determines the speed of the ball as it reaches the other side of the court.

How much force is applied to the ball via the *racket also affects* its velocity. How quickly the ball leaves your side of the court is determined by how quickly the ball gains its velocity.

To achieve a high-quality serve, a player has to gain as much speed on the ball as possible while maintaining control of its direction. A player can *either serve fast* or difficult to return due to spin on the ball, or both!

There are *many different types* of serves that can be mastered with time and practice. This article will discuss some of those serves and how to do them.

## Calculate the mass of both the ball and racket

In order to calculate the change in momentum of the ball, you need to know how much momentum was transferred from racket to ball.

You also need to know the mass of both the ball and racket in order to calculate this. The more mass a player has, the more momentum they can transfer to the ball.

The less mass a player has, the less momentum they can transfer to the ball. More mass players can create more impulse on the ball, but *less mass players* have more speed on their swings which can *also increase impulse* on the ball.

Mass can be determined by weighing someone or something and using that value in an equation that compares that number with weight and height. That gives you a *pretty good approximation* of **total body mass**.

## Calculate the momentum of both the ball and racket

Once you have calculated the velocity of the ball and racket, you can calculate the momentum of both objects. Momentum is defined as the quantity of motion of an object, and is denoted by p.

It is defined as *mass times velocity*, so that p=mv. Mass is what makes up the object, so if you know the mass, you can calculate the momentum!

To calculate the ball’s momentum, take its mass (mB) and subtract its displacement (yB). To calculate the racket’s momentum, take its mass (mR) and subtract its displacement (yR). Then, add these *two masses together* to get the *total mass involved* in this scenario.

Once you have determined all of these quantities, you can solve for their sum by dividing one by the other.

## Calculate the velocity of impact using Newton’s law of motion

Now that you know how to calculate the amount of time the ball was in the air, you can apply this knowledge to figure out how *much time* the ball was in contact with the racket.

You can do this by calculating the velocity of impact between the ball and the racket. This is done by dividing the velocity of the ball at impact by the length of time during which it was in contact with the racket.

If, for example, the **ball leaves** the racket moving at 10 m/s, and it *takes 0*.*5 seconds* for it to leave the racket, then its velocity of impact is 5 m/s. This means that an impulse of 5 m/s was applied to it by the racket.

Remember: An impulse is simply a change in momentum.

## Determine if there was a net force on the ball after collision

Now that you know how much momentum the ball received and how much momentum the racket lost, you can determine if there was a net force on the ball after collision.

If the ball lost momentum, then it had a negative net force. If the ball gained momentum, then it had a **positive net force**. If there was no change in momentum, then the net force was zero.

Imagine trying to balance a bookshelf on your head: if you move forward, you’ll fall down; if you move backward, you’ll fall down; but if you stay still, you’ll stay still. You **cannot stay balanced without** a * net force acting* on you.

The same goes for balls: if there is no net force acting on it, it will not move.

## Apply conservation of momentum to determine what impulse was applied to the ball by the racket

Now that you know how to find the velocity of the ball after the bounce, you can use that information in conjunction with conservation of momentum to determine what impulse was applied to the ball by the racket.

Conservation of momentum states that if no **external forces act upon** a system, its momenta will not change. In other words, the ball will retain its momentum before and after the bounce.

So how much force did the racket apply to the ball? You could experiment with balls of **different masses** and find out, but there’s an **easier way**.

Convert both objects to kilograms, then use this formula: Force = Mass x Acceleration.

## Check your calculations against Newton’s second law of motion

Once you have your answer, you should check it against Newton’s second law of motion. This law states that a body remains at rest or in *uniform motion unless* an external force is applied to it.

If the ball was moving before the serve, then your answer should account for the fact that the ball was moving and has a certain velocity before the serve impact. If the ball was at rest before the serve, then your answer should account for the fact that no external force was applied to the ball to make it move.

You can do this by comparing your answer to what *would happen* if there was no tennis racket—if no external force was applied to the ball, it **would remain** at rest. Your answer should compare similarly to this scenario.

## Check your calculations against energy conservation

Once you’ve calculated the ball’s velocity, you can calculate the ball’s **initial kinetic energy using** the formula:

Kinetic energy (of the ball) = ½mv²

Where m is mass (weight) of the ball, and v is velocity (speed) of the ball. You can then check your calculation against **energy conservation**: If you calculate how **much kinetic energy** the racket must have taken off the ball, plus how much it leaves in the ball’s speed, they should be equal!

If one of these values is higher than the other, then you probably made a mistake somewhere. Check your math and re-do any steps to fix them. Also, make sure that when you calculated the ball’s velocity that you used its final velocity, not its current velocity.

## What are some limitations to this analysis?

While this analysis is an excellent start, there are some limitations to this analysis. The first is that it assumes the ball was in rest position at the beginning of the swing.

As mentioned above, this is a reasonable assumption to make as the **ball typically starts** from rest. However, if the ball was moving at any velocity before the swing began, then this analysis would not be correct.

The second limitation is that it assumes no additional impulse is applied to the ball other than what the racket provides. If there was some friction between the ball and floor, or the ball bounced off of something after being hit, this analysis would not be correct.

The **third limitation** is that it assumes no elastic properties of the ball or racket. If either of these items were elastic, then there would be a *temporary increase* in velocity due to *internal forces countering one another*.