The Physics of Power: How Angular Momentum Shapes the Tennis Forehand

[airforce1]

In tennis...speed is one element of control, which is power. The other two are spin and placement.

great subject comment, but don't you mean speed is one of 3 elements of a "placement" with the 3 being speed, spin and direction?

[airforce1]

Here are a few thoughts:

In tennis, speed is power, and what matters most is racket head speed at the point of contact, not just raw mechanical force.

When a player like Roger Federer extends the arm, it doesn’t necessarily increase angular velocity. In fact, it can reduce it—because extending the arm increases the moment of inertia, or the resistance to rotation. However, it can increase total angular momentum, which is what ultimately matters when transferring energy through the kinetic chain.

This concept comes from the equation:

L = I x w

In this equation:
• L is angular momentum
• I is moment of inertia
• w (omega) is angular velocity, or the rate of rotation in radians per second

Even though omega may decrease due to the increased length of the lever, I (moment of inertia) increases substantially as mass is distributed farther from the axis of rotation (e.g., the shoulder). This can result in greater overall angular momentum, which helps generate high racket head speed when efficiently transferred through the kinetic chain.

Biomechanics researcher Dr. Brian Gordon has also hypothesized that the straight-arm forehand configuration—where the hitting arm is extended beyond 155 degrees—can contribute to producing a heavier ball. The reasoning is that this configuration allows for a greater contribution from internal shoulder rotation (ISR), which is the dominant source of racket head speed in the modern forehand. By maximizing ISR contribution, players may increase spin rate and momentum transfer to the ball, resulting in a shot that combines pace and heavy topspin.

This becomes especially effective when the full kinetic chain—starting from the ground and moving through the hips, torso, shoulder, arm, and wrist—is sequenced correctly. The result is a whip-like effect that generates maximum speed at the point of contact, even without an increase in torque at the shoulder.

Let’s continue the discussion—happy to explore the implications of ISR, angular momentum, and forehand configurations in more detail.

So my observation is that great players are extending their arm for the forehand to generate more angular momentum but also to generate higher spin rates.

This above was from CL, posted on Advanced Tennis Foundation. Here is a reply from one of the more educated Tennis Instructors in the US, Steve Kaplan doctorate in Bio-mechanics. You didn't reply there, but maybe you can address the mistakes he pointed out in his reply here?

Chris Lewit thank you Chris. Unfortunately "speed is not power" and racket head speed which is the denominator in the equation for power is no more significant than acceleration as a function of force which is the numerator in the equation for power. This is a common mistake. So the greatest power yield is the conversion of speed and acceleration at the apex of the x and y curve. I think you are aware as you recognize that third class levers sacrifice force for speed. I strongly disagree with Dr. Gordon . First internal shoulder rotation is not the dominant source of speed.It is the dominant transmitter of speed . Second, the contribution of this rotation is impacted by the arm configuration as a percentage but not as a raw number. This is also not well explained by Gordon and is a source of much confusion.Finally if you watch Roger while he times the reduction of his wrist extension very close to contact to increase monetary velocity to contact his ISR before impact is actually minimal as compared to players who flex their elbow sooner . It might be that the timing of this rotation is more effective but it is absolutely not greater in magnitude.

L= I x w tells us that a faster spinning object with a larger moment of inertia will have a greater moment of inertia and have a greater angular momentum. The moment of inertia is not " the resistance to rotation" as you said. It's resistance to rotational change. Angular momentum is the measure of an object's tendency to keep rotating.While such momentum is important to ensure that the energy created is transferred, it is not always conserved while swinging a racket and even more significant it is absolutely not a measure of power or energy transferred.. Power or the rate at which work is done is a different physical quality from angular momentum with different measures and applications. I'm sorry to tell you Chris that your formula while correct is very far off from having validity in the context of hitting a tennis ball with greater energy from an extended arm

Increasing the lever arm does not increase angular momentum in a third class lever. It can increase the angular momentum of a load that is being acted upon. The increased moment of inertia that you talk about is not an advantage here. In fact the increased moment of inertia offsets the potential increase in angular velocity which leads to the same or even lower angular momentum when compared to a shorter lever. In performing work, speed is a function of the time it takes to perform work the higher the speed the faster the work is performed, the greater the power. Equally important however is force which is a constant here in this example (mass) and a variable (acceleration). As explained power is work over time. Work is force x displacement. So for power, force is the top and speed is the bottom of this equation. What might be tripping you up is a common misunderstanding of "displacement" which in physics is not the total length traveled . That is distance. Displacement is the shortest distance between the starting and ending points. Your application of angular momentum , is as explained way off . Now as to spin rate; without greater power imparted by an extended arm which was derived here as a given, you can gain greater spin rate but at the cost of ball speed. Such a trade off exists with or without an extended arm or the magnitude of ISR because force vectors in swinging a racket can come from a multitude of directions ie. The shoulder rotation, the shoulder pitch angle relative to body and ball spacing, the elbow flex and orientation of rotation, the wrist extension and deviation, the body tilt from the pelvis , the knee flex and the ankle/shin angle. To pick ISR as the most significant factor for optimizing ball rotation is unproven by a correlation of a single joints contribution to energy transmission. Transmissions manage power flows, they do not create power.

[airforce1]

It’s not about having the ‘biggest’ it's about ‘efficiency’. It’s about a player adopting a functional backswing versus a positional backswing. The two best forehands ever thus far are Rafa’s and Roger’s – both positional. A functional backswing is going to involve more work to produce the same thing.

and from seano....Stotty -

I agree with your analysis about forehand efficiency. In addition, bent arm forehands require more effort because you need to bring momentum from the backswing into the forward swing to make up for coming from a closer set up position in the backswing. Positional backswings don't require any momentum from the backswing. Secondly, the use of the forearm is required for spin in a bent arm forehand There's not a direct line from the shoulder to the hand, like there is in a straight arm forehand, where the spin is created from the internal rotation of the shoulder (ISR). The bent arm requires ISR plus the use of the forearm to create the spin.

Stotty... I first addressed the fastest recorded Fhs being bent because that is the most common claim encountered from straight arm promoters, who try to say the most powerful Fhs are Straight armed. Good that you accept this common mistake is not an issue against bent arm Fhs.

As to the efficiency aspect, I thought that was clearly settled long ago, that with 3rd class levers such as the Fh, that the straight arm, with its longer radius/Lever is demonstrably LESS efficient than the shorter bent arm lever. Maybe there is a special type of efficiency you are addressing?

A straight line is not required for ISR and it seems now that no straight line is required even to have a straight arm, since the definition has been expanded by some to include many of the optimized "bent arm Fhs" that have around 155 degrees. Looks to me that since the evidence didn't support the straight arm Fh myths, that someone had to grab some of the best bent arm Fhs and redefine them as straight (despite them being bent).

Claiming that Fed and Nadal have the best ever 2 Fhs ever is so subjective that I don't see how it has a place in any serious discussion (unless there is some objective data to support it). Since we agreed that straight arms don't create the fastest Fhs, hopefully we can agree that they are not the most versatile as well... so hard to see why anyone would see these 2 straight arm Fhs as the 2 best ever.

I'm very interested in this functional vs positional backswing classification, as it is new to me.... and the odd naming, given that all backswings are both functional and positional by nature. There is no standard backwing that can't be done with both straight or bent Fhs (in contrast to the claim above). I'm guessing this topic comes from some article you can refer me to so I can review it.
I'd appreciate that greatly!
thanks

[jeremy93]

I would definitely guess the closed stance creates more power on the 1hbh. More rotational energy. Not surprised pros said neural as that is what a non-coach would most likely say.

Great post and thanks for your input. There are lot of different opinions over this one. That’s why I put it out there.

My argument would be a closed stance involves placing more weight on the back foot, which makes it harder to transfer energy efficiently forward into the shot. With a neutral stance, a player can push off both legs more equally and use the ground to drive upward and forward through the kinetic chain. It’s always going to flow better and allows the hips and torso to uncoil more freely toward the net. And, of course, there is neutral and less neutral and a whole range of stances between.

[jeremy93]

Stotty... I first addressed the fastest recorded Fhs being bent because that is the most common claim encountered from straight arm promoters, who try to say the most powerful Fhs are Straight armed. Good that you accept this common mistake is not an issue against bent arm Fhs.

As to the efficiency aspect, I thought that was clearly settled long ago, that with 3rd class levers such as the Fh, that the straight arm, with its longer radius/Lever is demonstrably LESS efficient than the shorter bent arm lever. Maybe there is a special type of efficiency you are addressing?

A straight line is not required for ISR and it seems now that no straight line is required even to have a straight arm, since the definition has been expanded by some to include many of the optimized "bent arm Fhs" that have around 155 degrees. Looks to me that since the evidence didn't support the straight arm Fh myths, that someone had to grab some of the best bent arm Fhs and redefine them as straight (despite them being bent).

Claiming that Fed and Nadal have the best ever 2 Fhs ever is so subjective that I don't see how it has a place in any serious discussion (unless there is some objective data to support it). Since we agreed that straight arms don't create the fastest Fhs, hopefully we can agree that they are not the most versatile as well... so hard to see why anyone would see these 2 straight arm Fhs as the 2 best ever.

I'm very interested in this functional vs positional backswing classification, as it is new to me.... and the odd naming, given that all backswings are both functional and positional by nature. There is no standard backwing that can't be done with both straight or bent Fhs (in contrast to the claim above). I'm guessing this topic comes from some article you can refer me to so I can review it.
I'd appreciate that greatly!
thanks

The fastest forehands ever recorded are usually bent arm because about 96% of pros use bent arm…..

[jeremy93]

Stotty... I first addressed the fastest recorded Fhs being bent because that is the most common claim encountered from straight arm promoters, who try to say the most powerful Fhs are Straight armed. Good that you accept this common mistake is not an issue against bent arm Fhs.

As to the efficiency aspect, I thought that was clearly settled long ago, that with 3rd class levers such as the Fh, that the straight arm, with its longer radius/Lever is demonstrably LESS efficient than the shorter bent arm lever. Maybe there is a special type of efficiency you are addressing?

A straight line is not required for ISR and it seems now that no straight line is required even to have a straight arm, since the definition has been expanded by some to include many of the optimized "bent arm Fhs" that have around 155 degrees. Looks to me that since the evidence didn't support the straight arm Fh myths, that someone had to grab some of the best bent arm Fhs and redefine them as straight (despite them being bent).

Claiming that Fed and Nadal have the best ever 2 Fhs ever is so subjective that I don't see how it has a place in any serious discussion (unless there is some objective data to support it). Since we agreed that straight arms don't create the fastest Fhs, hopefully we can agree that they are not the most versatile as well... so hard to see why anyone would see these 2 straight arm Fhs as the 2 best ever.

I'm very interested in this functional vs positional backswing classification, as it is new to me.... and the odd naming, given that all backswings are both functional and positional by nature. There is no standard backwing that can't be done with both straight or bent Fhs (in contrast to the claim above). I'm guessing this topic comes from some article you can refer me to so I can review it.
I'd appreciate that greatly!
thanks

The fastest forehands recorded are usually double bends largely because about 96% of players use double bends…..

[seano]

What's the record speed for double-bend forehands? Alcaraz was timed at 112mph at French Open.

[airforce1]

The fastest forehands recorded are usually double bends largely because about 96% of players use double bends…..

You make an excellent point. A vast majority of the best players use bent arm Fhs and that Imo strongly suggest that the Bent arm is at least on par with the Straight arm Fh (my only claim) or more players would migrate to the Straight arm Fh as they have to the ATP Backswing and the CTM Swivel to the slot entry.

That aside, if the SA was a more powerful or better shot by physics, we would see faster Fhs from it even if it was a more avg player and surely from some of the top players like Fed.

Just a key reminder.... I'm not trying to hype the bent arm Fh as better.... I'm just pointing to how it is a mere style choice where there has been no solid physics to support that the SA is better for humans.... other than junk science and opinion.

[airforce1]

What's the record speed for double-bend forehands? Alcaraz was timed at 112mph at French Open.

Fed has also hit a 118 mph, so the straight arm is no slouch of a technique... but that there are several Bent Fhs recorded over 120 (I think the record is 125 currently) shows the bent arm Fh is very capable of major power and maybe even more than the SA Style Fh. They are both acceptable choices, so people should do what they like.... But Coaches shouldn't get it twisted and try to convince others of imagined superiority of the SA Style just because they are a fan of it.

[jimlosaltos]

Fed has also hit a 118 mph, so the straight arm is no slouch of a technique... but that there are several Bent Fhs recorded over 120 (I think the record is 125 currently) shows the bent arm Fh is very capable of major power and maybe even more than the SA Style Fh. They are both acceptable choices, so people should do what they like.... But Coaches shouldn't get it twisted and try to convince others of imagined superiority of the SA Style just because they are a fan of it.

I believe, from memory, James Blake hit the 125 MPH, and Delpo was in that vicinity. I remember a lot of things about James but not where to categorize his forehand.

[stotty]

Stotty... I first addressed the fastest recorded Fhs being bent because that is the most common claim encountered from straight arm promoters, who try to say the most powerful Fhs are Straight armed. Good that you accept this common mistake is not an issue against bent arm Fhs.

As to the efficiency aspect, I thought that was clearly settled long ago, that with 3rd class levers such as the Fh, that the straight arm, with its longer radius/Lever is demonstrably LESS efficient than the shorter bent arm lever. Maybe there is a special type of efficiency you are addressing?

A straight line is not required for ISR and it seems now that no straight line is required even to have a straight arm, since the definition has been expanded by some to include many of the optimized "bent arm Fhs" that have around 155 degrees. Looks to me that since the evidence didn't support the straight arm Fh myths, that someone had to grab some of the best bent arm Fhs and redefine them as straight (despite them being bent).

Claiming that Fed and Nadal have the best ever 2 Fhs ever is so subjective that I don't see how it has a place in any serious discussion (unless there is some objective data to support it). Since we agreed that straight arms don't create the fastest Fhs, hopefully we can agree that they are not the most versatile as well... so hard to see why anyone would see these 2 straight arm Fhs as the 2 best ever.

I'm very interested in this functional vs positional backswing classification, as it is new to me.... and the odd naming, given that all backswings are both functional and positional by nature. There is no standard backwing that can't be done with both straight or bent Fhs (in contrast to the claim above). I'm guessing this topic comes from some article you can refer me to so I can review it.
I'd appreciate that greatly!
thanks

Sorry for the delay, airforce1. You got buried under other posts!

The positional vs functional backswing debate is systematically explained by Brian Gordon in one of The Four Pillars videos. I’m not sure which video in the article but it’s there somewhere.

Yes, it’s subjective to call a shot "great" without data or evidence to back it up. I admit, it's just my opinion. But I’m entitled to it, and I’m sticking with it. I’ve never seen a forehand as extraordinary as Roger’s. I watched it one sunny afternoon at SW19 when he dismantled Milos Raonic. It was a majestic sight.

Raonic had beaten an injured Roger the year before, so this match was as much about setting the record straight as advancing to the next round. That day, Federer hit forehand after forehand—some with blistering pace, some loaded with spin, others placed with surgical precision in awkward spots. And he missed very few.

The powerful forehands were explosive, and they quite literally took the crowd’s breath away. Wimbledon crowds are a mix of the knowledgeable and the clueless—but when it comes to moments of shock and awe, everyone gets it…everyone certainly got it that day.

[jeffreycounts]

Sorry for the delay, airforce1. You got buried under other posts!

The positional vs functional backswing debate is systematically explained by Brian Gordon in one of The Four Pillars videos. I’m not sure which video in the article but it’s there somewhere.

Yes, it’s subjective to call a shot "great" without data or evidence to back it up. I admit, it's just my opinion. But I’m entitled to it, and I’m sticking with it. I’ve never seen a forehand as extraordinary as Roger’s. I watched it one sunny afternoon at SW19 when he dismantled Milos Raonic. It was a majestic sight.

Raonic had beaten an injured Roger the year before, so this match was as much about setting the record straight as advancing to the next round. That day, Federer hit forehand after forehand—some with blistering pace, some loaded with spin, others placed with surgical precision in awkward spots. And he missed very few.

The powerful forehands were explosive, and they quite literally took the crowd’s breath away. Wimbledon crowds are a mix of the knowledgeable and the clueless—but when it comes to moments of shock and awe, everyone gets it…everyone certainly got it that day.

I've read several tour players say that Del Potro had the biggest forehand. Personally, I like Roger's the best. So beautiful to watch while also being brutally lethal. I only saw Roger once live and it was in the upper stands, but it sounds like you were closer Stotty. I'm jealous!

[airforce1]

Sorry for the delay, airforce1. You got buried under other posts!

The positional vs functional backswing debate is systematically explained by Brian Gordon in one of The Four Pillars videos. I’m not sure which video in the article but it’s there somewhere.

Yes, it’s subjective to call a shot "great" without data or evidence to back it up. I admit, it's just my opinion. But I’m entitled to it, and I’m sticking with it. I’ve never seen a forehand as extraordinary as Roger’s. I watched it one sunny afternoon at SW19 when he dismantled Milos Raonic. It was a majestic sight.

Raonic had beaten an injured Roger the year before, so this match was as much about setting the record straight as advancing to the next round. That day, Federer hit forehand after forehand—some with blistering pace, some loaded with spin, others placed with surgical precision in awkward spots. And he missed very few.

The powerful forehands were explosive, and they quite literally took the crowd’s breath away. Wimbledon crowds are a mix of the knowledgeable and the clueless—but when it comes to moments of shock and awe, everyone gets it…everyone certainly got it that day.

Thank you for the reference to the 4 Pillars... and yes, you are very entitled to your studied opinion on Quality!