Netzroller
Semi-Pro
There is a lot of discussion going on about how racket weight affects power.
Well, I tried to do some calculations to get a more accurate answer to this question.
Here's what I did:
-Let's model a tennis shot as an ideal elastic collision of two object - racket and ball.
As you know, swingweight considers the mass distribution of a racket. This becomes important, when the racket is swung in a circular way. The further the mass is from the center of rotation, the heavier the racket appears to be. (This also shows that swingweight is a rather arbitrary number, since it depends on how exactly the racket is swung).
However, we assume the swing path to be a straight line → swingweight = actual weight.
Since I think the ball has left the racket long before the player can counteract the impact, it seems as if arm and racket are not connected. Only the racket recoils in the moment of impact, not the whole arm (this happens after the ball has already left the stringbed). Therefore, the weight of the player/arm of the player becomes irrelevant.
In an ideal elastic collision, no kinetic energy is lost. As opposed to an inelastic collision where the objects are deformed (energy loss in form of heat) and stick together after the collision.
For a tennis shot, we have an in between case, since there is energy loss mainly due to ball deformation.
However, this loss will occur for both, shots with light and heavy rackets pretty much equally. Since we are mainly interested in relative differences and I don't know the exact behavior of racket and ball during collision we treat this as an perfectly elastic collision.
This means:
conservation of impulse: m1*v1+m2*v2 = m1*v1'+m2*v2'
conservation of energy: 0.5m1*v1^2+0.5m2*v2^2= 0.5m1*v1'^2+0.5m2*v2'^2
m1 = racket mass
v1= swing velocity
m2 = ball mass
v2= ball velocity (has the opposite sign of v1)
the v' indicates the speed after the collision.
Solving the equations gives us:
v1' = 2(m1*v1+m2*v2)/(m1+m2) -v1
v2' = 2(m1*v1+m2*v2)/(m1+m2) -v2 (this is what we focus on since it gives us how fast out shot will be)
(note 1:
In reality, the equation would be v2'= (m1*v1+m2*v2-m1(v2-v1)k)/(m1+m2)
k describing the coefficient of restitution (k=1 perfectly elastic; k=0 perfectly inelastic), which depends on various factors like strung weight, frame stiffness etc.
note2:
I used lots of parenthesis and * signs so you will all know what I mean - whatever you voted for in the “48/2(9+3)” thread:twisted: )
Now we are interested in how a change in racket weight and swing speed affects v2'. This can be done using the partial derivative:
racket weight change:
d/(d(m1))(v2')=2*v1/(m1+m2) -2*m1*v1/(m1+m2)^2=...=2*v1*m2 / (m1+m2)^2
swing speed change:
d/(d(v1))(v2')=2*m1/(m1+m2)
This result is crucial!!!
Swing speed has a linear effect in power (there is no v1 dependency). How great the effect of additional swing speed is depends on the masses of racket and ball alone.
Therefore, swinging the racket x% faster will always result in x% more power. (in reality slight differences in ball deformation etc. will slightly change this).
However, the effect of racket weight is proportional to 1/m^2 ! This means, a heavier racket swung at the same speed does gives you more power. However, the heavier the racket already is, the less additional power you get from increasing the weight even further. There is actually a theoretical limit to where more weight wouldn’t have any more benefit: Lets say m2 approximates infinity. If we now use the equation for v2' we see the result approaches asymptotically 2*v1-v2.
To illustrate this, whether you throw a ball against a car or against a bus, it will come back at pretty much the same speed, even though the bus is much heavier.
On the other hand, swing speed always pays off. There are no boundaries, if you increase v1 further, v2' will get higher and higher.
Example:
m1 = 300g
v1= 10m/s
m2 = 60g
v2= -20m/s
->v2'= 30m/s
increasing weight by 10% (m1=330g) → v2'= 30.77m/s
increasing swing speed by 10% (v1=11m/s) → v2'= 31.67m/s
increasing weight by 50% (m1=450g) → v2'= 32.9m/s
increasing swing speed by 50% (v1=15m/s) → v2'= 38.3m/s
Comparing the two derivatives we also find one thing:
The effect of increased swing speed is constant whereas it starts at infinity and sharply drops for increased weight. Thus, there must be a distinct point, from which on increasing swing speed by x% is always better than adding x% more weight.
Looking at the equation for v2' we notice: The term “2(m1*v1+m2*v2)/(m1+m2)” is zero for p1<=>m1*v1=m2*v2<=>p2 (since v2 has the opposite sign). The higher the result is the better, because it means higher velocity. For p1<p2 the term is negative. Increasing m1 by a certain percentage is better than increasing v1 by that same percentage since the former will keep the negative term small. However, for p1>p2, the term is positive. You therefore want to make it big. In order to do that, increasing v1 is more effective than increasing m1.
Therefore: From the point on where the impulse of the racket exceeds the impulse of the ball, increase in swing speed by x% has a greater effect on increasing power than adding x% racket weight!
Given every racket is much heavier than the ball, this should be pretty much always true for fast strokes! (I don't know what racket head speeds are realistic for serves and ground stokes. For volleys or block returns the racket certainly has a low impulse, therefore high weight is better for power)
Therefore, swinging faster is usually more effective than adding weight - provided the increase is of the same percentage. If you can either for example swing 5% faster of add 20% more weight, things look differently and the result might shift in favor of adding weight.
This could explain why someone gets less power using lighter frames. He cannot increase his swing speed the same amount he lowered the racket weight.
However, this calculation also reveals several basic cons of using lighter frames.
First of all, the Kinetic Energy is a function of the mass and the velocity squared. In order to accelerate the racket, the body has to bring up that energy. Relying on high swing speed rather than on racket weight is therefore generally more exhausting. Over the course of a match, this might lead to a significant drop in power due to a drop of physical performance.
If we shift our attention to the equation for v1' (racket after impact), the same pattern occurs: From the same distinct point on, faster swinging will lead to a greater recoil of the racket than adding weight. This may not be desired, since it leads to impact shock and vibration. If the ball is hit not clean, this point might be of even greater importance.
On the other hand, if the racket is too heavy it might temper your ability to execute your shots properly and prepare in time. Thus, leading to less power, and probably more discomfort.
Conclusion:
Heavier rackets provide more power. However, increasing swing speed by the same amount has usually a more significant effect. Therefore, if a light racket allows you to swing faster, you will be able to hit harder.
Unfortunately, these equations can still not tell you, what racket you should use.
In order to draw more conclusion from this, one would need to know more on the bio-mechanical background: Can the body rather move heavier things slow or light things fast? What feels more comfortable and which speed can be coordinated best? How does performance drop over the course of a match? At what point does racket weight get in the way of your mechanics?
This calculation solely focuses on peak power. Other important aspects like comfort, feel and control are neglected.
Well, I don't know if anyone is going to read such a long text of if you think it's a whole bunch of crap
Well, I'm a physics student who loves tennis so I had a lot of fun thinking about the problem anyways. I thought I might just as well share my thoughts with you...
Any corrections, comments or discussions are highly appreciated!
Well, I tried to do some calculations to get a more accurate answer to this question.
Here's what I did:
-Let's model a tennis shot as an ideal elastic collision of two object - racket and ball.
As you know, swingweight considers the mass distribution of a racket. This becomes important, when the racket is swung in a circular way. The further the mass is from the center of rotation, the heavier the racket appears to be. (This also shows that swingweight is a rather arbitrary number, since it depends on how exactly the racket is swung).
However, we assume the swing path to be a straight line → swingweight = actual weight.
Since I think the ball has left the racket long before the player can counteract the impact, it seems as if arm and racket are not connected. Only the racket recoils in the moment of impact, not the whole arm (this happens after the ball has already left the stringbed). Therefore, the weight of the player/arm of the player becomes irrelevant.
In an ideal elastic collision, no kinetic energy is lost. As opposed to an inelastic collision where the objects are deformed (energy loss in form of heat) and stick together after the collision.
For a tennis shot, we have an in between case, since there is energy loss mainly due to ball deformation.
However, this loss will occur for both, shots with light and heavy rackets pretty much equally. Since we are mainly interested in relative differences and I don't know the exact behavior of racket and ball during collision we treat this as an perfectly elastic collision.
This means:
conservation of impulse: m1*v1+m2*v2 = m1*v1'+m2*v2'
conservation of energy: 0.5m1*v1^2+0.5m2*v2^2= 0.5m1*v1'^2+0.5m2*v2'^2
m1 = racket mass
v1= swing velocity
m2 = ball mass
v2= ball velocity (has the opposite sign of v1)
the v' indicates the speed after the collision.
Solving the equations gives us:
v1' = 2(m1*v1+m2*v2)/(m1+m2) -v1
v2' = 2(m1*v1+m2*v2)/(m1+m2) -v2 (this is what we focus on since it gives us how fast out shot will be)
(note 1:
In reality, the equation would be v2'= (m1*v1+m2*v2-m1(v2-v1)k)/(m1+m2)
k describing the coefficient of restitution (k=1 perfectly elastic; k=0 perfectly inelastic), which depends on various factors like strung weight, frame stiffness etc.
note2:
I used lots of parenthesis and * signs so you will all know what I mean - whatever you voted for in the “48/2(9+3)” thread:twisted: )
Now we are interested in how a change in racket weight and swing speed affects v2'. This can be done using the partial derivative:
racket weight change:
d/(d(m1))(v2')=2*v1/(m1+m2) -2*m1*v1/(m1+m2)^2=...=2*v1*m2 / (m1+m2)^2
swing speed change:
d/(d(v1))(v2')=2*m1/(m1+m2)
This result is crucial!!!
Swing speed has a linear effect in power (there is no v1 dependency). How great the effect of additional swing speed is depends on the masses of racket and ball alone.
Therefore, swinging the racket x% faster will always result in x% more power. (in reality slight differences in ball deformation etc. will slightly change this).
However, the effect of racket weight is proportional to 1/m^2 ! This means, a heavier racket swung at the same speed does gives you more power. However, the heavier the racket already is, the less additional power you get from increasing the weight even further. There is actually a theoretical limit to where more weight wouldn’t have any more benefit: Lets say m2 approximates infinity. If we now use the equation for v2' we see the result approaches asymptotically 2*v1-v2.
To illustrate this, whether you throw a ball against a car or against a bus, it will come back at pretty much the same speed, even though the bus is much heavier.
On the other hand, swing speed always pays off. There are no boundaries, if you increase v1 further, v2' will get higher and higher.
Example:
m1 = 300g
v1= 10m/s
m2 = 60g
v2= -20m/s
->v2'= 30m/s
increasing weight by 10% (m1=330g) → v2'= 30.77m/s
increasing swing speed by 10% (v1=11m/s) → v2'= 31.67m/s
increasing weight by 50% (m1=450g) → v2'= 32.9m/s
increasing swing speed by 50% (v1=15m/s) → v2'= 38.3m/s
Comparing the two derivatives we also find one thing:
The effect of increased swing speed is constant whereas it starts at infinity and sharply drops for increased weight. Thus, there must be a distinct point, from which on increasing swing speed by x% is always better than adding x% more weight.
Looking at the equation for v2' we notice: The term “2(m1*v1+m2*v2)/(m1+m2)” is zero for p1<=>m1*v1=m2*v2<=>p2 (since v2 has the opposite sign). The higher the result is the better, because it means higher velocity. For p1<p2 the term is negative. Increasing m1 by a certain percentage is better than increasing v1 by that same percentage since the former will keep the negative term small. However, for p1>p2, the term is positive. You therefore want to make it big. In order to do that, increasing v1 is more effective than increasing m1.
Therefore: From the point on where the impulse of the racket exceeds the impulse of the ball, increase in swing speed by x% has a greater effect on increasing power than adding x% racket weight!
Given every racket is much heavier than the ball, this should be pretty much always true for fast strokes! (I don't know what racket head speeds are realistic for serves and ground stokes. For volleys or block returns the racket certainly has a low impulse, therefore high weight is better for power)
Therefore, swinging faster is usually more effective than adding weight - provided the increase is of the same percentage. If you can either for example swing 5% faster of add 20% more weight, things look differently and the result might shift in favor of adding weight.
This could explain why someone gets less power using lighter frames. He cannot increase his swing speed the same amount he lowered the racket weight.
However, this calculation also reveals several basic cons of using lighter frames.
First of all, the Kinetic Energy is a function of the mass and the velocity squared. In order to accelerate the racket, the body has to bring up that energy. Relying on high swing speed rather than on racket weight is therefore generally more exhausting. Over the course of a match, this might lead to a significant drop in power due to a drop of physical performance.
If we shift our attention to the equation for v1' (racket after impact), the same pattern occurs: From the same distinct point on, faster swinging will lead to a greater recoil of the racket than adding weight. This may not be desired, since it leads to impact shock and vibration. If the ball is hit not clean, this point might be of even greater importance.
On the other hand, if the racket is too heavy it might temper your ability to execute your shots properly and prepare in time. Thus, leading to less power, and probably more discomfort.
Conclusion:
Heavier rackets provide more power. However, increasing swing speed by the same amount has usually a more significant effect. Therefore, if a light racket allows you to swing faster, you will be able to hit harder.
Unfortunately, these equations can still not tell you, what racket you should use.
In order to draw more conclusion from this, one would need to know more on the bio-mechanical background: Can the body rather move heavier things slow or light things fast? What feels more comfortable and which speed can be coordinated best? How does performance drop over the course of a match? At what point does racket weight get in the way of your mechanics?
This calculation solely focuses on peak power. Other important aspects like comfort, feel and control are neglected.
Well, I don't know if anyone is going to read such a long text of if you think it's a whole bunch of crap
Well, I'm a physics student who loves tennis so I had a lot of fun thinking about the problem anyways. I thought I might just as well share my thoughts with you...
Any corrections, comments or discussions are highly appreciated!
