# Serve Speed Revisited - My 118mph Serve (using physics + accounting for drag)

Discussion in 'Tennis Tips/Instruction' started by ximian, Oct 25, 2009.

1. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
So I did a quick shoot of my serve from four different angles while trying out something slightly new in my motion. On a couple of my serves, you can calculate how many frames the ball takes from racket to court, so I decided to calculate my serve speed. Here's how it went down:

Here's my serve:

On the first serve, I counted 12 frames from the point of impact to touching the service line. The camera shoots at 30 fps, so it took .4 seconds to travel that distance.

dt = .4 seconds

Next, I need to calculate what that distance was. The ball hit the service line, almost halfway (about 40%) between the T and out wide. Using the dimensions of a tennis court:

Length from baseline to service line: 18.29 meters
Length from T to out wide: 4.115 meters
40 % of T to out wide: 1.646

Next, I need to somehow figure out how high the point of impact was. On some of the side serves you can tell that I'm about 6" off the ground when I hit, but I'm hunched over. I took some quick measurements and roughly calculated that I'm about 9' up. Or 2.74 meters.

Now I can use Pythagoras and find that the total distance the ball traveled is
sqrt[(18.29^2)+(1.646^2)+(2.74^2)] = 18.57 meters

d=18.57 m

I know that these measurements are approximate, because I don't know the exact height of impact, I'm a little inside and to the left of the court when I strike the ball, and I'm making my best guess as to where the ball lands. But the fact is, a margin of error of .5 meters still won't make a big difference on the overall speed, so it's still going to be a pretty accurate measurement.

Next, I need to account for drag. Think about balling your hand up and sticking it out the window of a car moving at 100 mph, and you'll get a similar idea of what sort of force drag is putting on my serve. That's pretty significant.

The formula for drag, if the object is between a few mm and a few m, and it's traveling under 200 m/s near the earth's surface is:

D=(1/4)(A)(v^2)

where A is the cross section, and v is it's velocity. The cross section of a tennis ball is simply (pi)(r^2). So I need the radius of a tennis ball, which is roughly:

r=0.0333 m

The problem with drag is that I need to know it's velocity, which is what I'm trying to calculate in the first place. However, since this is going to be a rough estimate of the type of force drag will place on a tennis ball, I can calculate the velocity of the ball using just distance and time, add a few mph for good measure, and use that in this calculation. So velocity is distance traveled over time, or:

v= d/dt
v= 18.57/.4
v= 46.425 m/s or 103.9 mph

Add a few m/s for a rough factor of drag, and I'll use 50 m/s (111.8 mph) as the velocity in my Drag equation. So:

D = (1/4)(pi)(r^2)(v^2)
D = 2.18

I'll leave the units of D out since that's a little complex. But I do know that for the drag equation to work, you need all your units in kg, m, and s, which I did.

Ok, so now that I have my force from Drag, I can take another look at the flight of this tennis ball.

If you draw a force diagram (a picture of all the forces acting on the ball) of the tennis ball right after it's struck, you'll notice just two forces (there are others, but they're negligible):

Gravity, or F(g)
Drag, or D

But the flight of the ball is not perfectly horizontal. It's heading down from a height of 2.74 m, so the axis is slightly tilted. Note that F(g) is pointing straight down, while D will oppose the motion of the ball, traveling slightly down. Thus, using Newton's second law (F=ma) and since we are only interested in the forces that oppose motion (x-axis), then we can calculate all the forces in the x-axis find the ball's acceleration. We expect it to be negative because the ball is slowing down due to drag.

F(g) + D = ma

where:
F(g) = mg
D = -2.18 (negative because it opposes motion)
m = 0.0567 kg (mass of tennis ball)
g = 9.8 m/s^2
a = acceleration

But let's take another look at gravity. Gravity is pointing down, so most of it's force is in the y-direction. But since the axis is tilted, there will be a small amount of gravity that is actually speeding the ball up. This component is equal to sin(theta) where (theta) is the angle measured from the court up to the point of impact.

(theta) = sin^(-1) [2.74/18.57]
(theta) = 8.5 degrees

So the final equation for acceleration is:

a = [sin(theta)mg+D]/m
a = -37.00 m/s^2

This seems a bit high doesn't it? But think back to the window and the car. Sticking your hand out the window and you'll definitely feel a strong force. That force is big enough to cause a deceleration like this. Plus, as we'll see in a bit, this acceleration force ends up looking much more rational when it's plugged into the proper equation.

So now that we have acceleration, distance, and time, we can finally calculate it's initial velocity. Newton's laws give rise to Kinematic equations, one of which is:

P(final) = P(initial)+(v)(dt^2)+(1/2)(a)(dt^2)

We have all the data except v, so a quick number crunch gives:

18.57 m = 0 + v (.4^2) - 2.96 m

The acceleration term calculates to just under 3 meters. Or in other words, the force of drag means that the ball traveled 3 meters less than it would have without drag. This seems reasonable. Solving for v gives:

v = 53.825 m/s or 120.4 mph

But let's look at our assumptions. I assumed that the height was 9' when I struck the ball, and that dt is .4 seconds, and that the mass and the radius were the numbers given (found from this website: http://hypertextbook.com/facts/2000/ShefiuAzeez.shtml). These are all reasonable, but just to be safe let's take .5 m off the distance traveled and recalculate. In that case, it's still 52.575 m/s or 117.6 mph. Still in the same ballpark. The other terms are relatively negligible as well, so I'll conservatively say that first serve was probably at least 118 mph, possibly as high as 121.

Additionally, I can calculate the difference between initial velocity and final velocity, using this equation:

v(final) = v(initial)+(a)(dt)
using a conservative 118 mph (52.57 m/s) as my initial velocity, I get:
v(final) = 37.77 m/s or 84.5 mph

This seems about right. The ball is certainly not traveling even 85 mph when the returner hits the ball, and we know that the impact between the ball and the court will slow it down more. These results look right.

I must admit I'm a little surprised it measured that fast. I was expecting something around 110, considering I wasn't trying to hit it really hard, and the tennis balls were pretty flat... But it is what it is. Now I'm curious what I can measure with new strings / balls / and trying to for a really big one!

Thoughts?

Last edited: Oct 25, 2009
2. ### ubermeyerHall of Fame

Joined:
Mar 1, 2009
Messages:
2,176
Location:
Texas
not sure I fully understand this... let's say you were a pro, would the radar gun in the standard place for pro matches measure it as 85 mph or 120 mph?

3. ### XenakisHall of Fame

Joined:
Sep 16, 2009
Messages:
2,424
Nice serve, well shot too (the video I mean.)

Just a side point, are you aware of this online tool for measuring serve speed from video footage?, might be interesting to compare results if you haven't tried already.

http://www.donthireddy.us/tennis/speed.html

12 frames to travel 60 feet (ish, you are slightly to the right of the T but you hit from slightly inside the court) at NTSC rate (29.97fps) equals 120mph (initial speed) using that online tool, looks about right to me and matches your figure more or less (don't ask me about the maths/physics, not my topic.)

Anyway, good work whether or not the figure is exact to 1mph or whatever.

Last edited: Oct 25, 2009
4. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
I think on tour they point the radar guns right at the server's racket, so that should measure 120.

5. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
wow that's a pretty cool website. Nice to see my figures match up with theirs as well.

Joined:
Sep 3, 2004
Messages:
1,626
7. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
44,842
You are hitting the ball well into the court, so the ball has actually traveled a shorter distance, and the speed would be less than your calculations.

8. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
Mike, did you read my post? even if you take off .5 m, it's still 117 mph so it's not as significant as you might think.

9. ### Mike CottrillHall of Fame

Joined:
Sep 3, 2004
Messages:
1,626
Look at my example here: http://www.donthireddy.us/tennis/RoddickExample.html
• 58.84’ == 139.04mph
• 58.50’ == 138.10mph
• 58.25’ == 137.41mph
• 58.00’ == 136.73mph
Notice the difference between 58.84’ and 58.50’ is only 4”.

I guess it depends on how much error you consider significant.

10. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
44,842
Thank you very much for this insight - you are hitting DOWN on the ball at impact, and you are not Karlovic.

You do not know how many insults I had to endure on this board for saying that most pro first serves are hit down, not up. This is specially obvious on flat serves. The server reaches up to the ball, but when the ball leaves the strings after the dwell time, and after the pronation or snap whatever you call it, it often is headed down. Sometimes the ball is caught on the strings slightly before maximum racquet height, dwells on it thru the crossover from up to low, and then leaves the strings heading downwards. For second kick serves, the ball is often really hit up on impact, but even for these serves, for a small percentage of them, the impact is a roll over the ball, pushing it down.

11. ### Mike CottrillHall of Fame

Joined:
Sep 3, 2004
Messages:
1,626
^^
Lighten up.. you are reading to much into what he wrote. This is about calculating a serve speed not technique. The guy knows how to hit a serve..

12. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
44,842
Of course he does. That is why I was interested.

13. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
Actually sureshs has a point, and it's something I overlooked. The flight of the ball is not linear, it's more parabolic, and it should be pretty much straight horizontal at impact and then come down due to gravity and spin. I just slowed down on of the side serves so I could look at each frame, and sure enough the ball traveled in what looked like a completely horizontal trajectory.

This will change my equations a little. The force of acceleration no longer has the sin(theta)(m)(g) component, but that was so small it was almost negligible. Without that component, the acceleration calculates to:

a = -38.45 m/s^2

which affects the final velocity:

v = 54.115 m/s or 121.05 mph.

So it makes a difference, but not much.

Side Note: This will also change my final velocity, because now the sum of forces along the direction of motion are not constant. initially, theta = 0 degrees, and at impact it will probably be around 10 degrees. Thus the acceleration component is not linear, and that creates a whole new ball of wax. But the difference will (again) be pretty negligible.

Last edited: Oct 25, 2009
14. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
Exactly. A few mph is not significant for me, I'm looking for ballpark. There are way too many assumptions for me to pinpoint the speed with any definite accuracy anyway. I'm just content that its in the ballpark of 120 mph.

15. ### Mike CottrillHall of Fame

Joined:
Sep 3, 2004
Messages:
1,626
BTW,

If you join your serves like:

or
http://www.vimeo.com/2310730

It will give you an interesting perspective stepping frame by frame.

16. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
44,842
That is correct - it is more like a parabola, but not exactly one. It would be exactly a parabola if air drag was not present. The ball leaves at horizontal or a couple of degrees below it, and follows a parabola-like path with the bulge of the curve above the net. That is why the similar triangle calculations which attempt to prove that serves cannot be hit down unless you are King Kong are wrong - they assume a straight line from racquet to touchdown. Yet the very same people drawing the straight line going down will also claim that the ball is headed up after impact!!! At least if they said it was parabolic, they would leave room for themselves to say it is hit up - as it is on many kick serves.

It is interesting that not even all kick serves are hit up. I used to think so, but couple of things changed my mind. One was an observation from a commentator that a particular pro kick serve only cleared the net by a few inches. Didn't seem likely to have been hit up. Another was a comment by Lindsay Davenport (commentating) that someone's kick serve was hit over the ball. There is a variant of the kick serve which is hit towards the top of the ball downwards, and not from low to high side of the ball.

17. ### ubermeyerHall of Fame

Joined:
Mar 1, 2009
Messages:
2,176
Location:
Texas
Are you a kid? No offense meant at all, but you look pretty young.

Oh and... I don't know, that doesn't look 110-120 mph. I can't argue with your math, but for some reason it just doesn't look that fast.

18. ### Mike CottrillHall of Fame

Joined:
Sep 3, 2004
Messages:
1,626
All good points, but I still believe with all these assumptions, high speed video and accurate distance, video + the speed calc provides more accurate readings than the radar guns. I'm surprised tennis broadcast producers have not used shot spot velocities more in their tennis productions.

19. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
I just crunched some numbers, and it's changing my perspective on the trajectory of a tennis ball. If someone hits a perfectly flat tennis ball at 50 m/s (about 112 mph) perfectly horizontal, the ball will only fall about .4 meters when it crosses the net. That's not a lot.

However, real life is not like this. The ball slows down significantly as it travels, and there is at least some spin imparted on the ball. This spin creates a low pressure zone and adds to the downward force, and the deceleration causes a longer time period. But even factoring in these changes, the ball can still probably be struck down at a slight angle, clear the net, and land in the court. I'm not positive, but that's where my hunch is leading. Interesting.

Another thing this will affect is the total distance the ball travels. Now we have to integrate over a curve to find d, since it is not linear. But this will only add to the distance traveled, and increase the final velocity accordingly.

20. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
I'm a college student. I get the young quote fairly often though, so it's fine.

I agree. It doesn't seem that fast. But the numbers don't lie, and they all seem accurate. The key here is to understand that the ball slows down significantly after it's struck. It's only traveling 70% of it's initial speed when it hits the court, and much less when it hits your racket.

21. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
44,842
The problem is it only works for hard and flat serves. Most pro serves have a lot of spin, of one kind or the other, and the path of the ball does not lie in one plane, as your calculation requires. The ball moves sideways in the air. Then you cannot apply this method.

22. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
No you can still apply this method. You just need to adjust for three dimensions instead of two, but the calculations will still be very accurate, and if you have the distance and the time perfectly calculated, it will be much more accurate than any radar gun.

23. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
44,842
Yes, but the calculations would not be easy at all. It would be simpler to just measure the impact speed with a gun.

24. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
Not if you know your physics

25. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
This looks pretty sweet, but I don't know how to do that. All I have is Window's Movie Maker.

26. ### Mike CottrillHall of Fame

Joined:
Sep 3, 2004
Messages:
1,626
If you have x, y and z, you got it. Shot spot provides it all.

27. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
44,842
You can do a big service to this board. How far behind the baseline was the fence? The standard distance of 21 feet or the tournament distance of 26 feet?

You should come up with a formula as a rule of thumb to calculate the speed from the height the ball hits the fence after one bounce on a standard court. It will be very helpful.

Joined:
Sep 3, 2004
Messages:
1,626
29. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
This won't work at all because the height of the bounce will vary greatly depending on how much spin you put on the ball. Some of my kickers can hit the same height, but they're not nearly as fast.

Joined:
Sep 3, 2004
Messages:
1,626
Not really.

31. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
44,842
That is why I said rule of thumb. A range would be enough.

32. ### Mr.BrightsideRookie

Joined:
Aug 12, 2009
Messages:
283
Location:
New York City
regardless of the speed, that was a nice serve man

33. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
Thanks! I'll take a look at that.

And no, that's not me. I wish though. Maybe I can get some match play up sometime soon.

No there would just be too much margin of error. The range would be anywhere from 130 to 100 or something like that. Big enough to be useless.

Thanks!

34. ### ubermeyerHall of Fame

Joined:
Mar 1, 2009
Messages:
2,176
Location:
Texas
That is impossible. I can serve an underhanded "Superlob" that lands in and after one bounce, if the player does not hit an overhead, can fly over the fence (20 feet or so back). I can also serve an overhead flat serve much faster than this (though still slow) that does not even hit the fence off one bounce. The problem is that, perhaps my underhand serve IS faster than my overhead serve in terms of total velocity, in other words, because it is also traveling rapidly vertically, while my flat serve has much less vertical movement. However, for all intents and purposes we don't care about vertical velocity, only horizontal velocity.

35. ### 5263G.O.A.T.

Joined:
Mar 31, 2008
Messages:
11,791
That is why the rule of thumb does not apply to kickers, and is used for the flatter first serve.
This is for serving, not some super lob Ubermeyer.
It's not hard to come up with some alternate method to things that will circumvent any rule of thumb. It's part of how magic is done, as your brain is often using rules of thumb that can be fooled.
In this case though, you only fool yourself, so not sure the sense in that.

And No, it is initial velocity we are concerned with.

Last edited: Oct 25, 2009
36. ### 5263G.O.A.T.

Joined:
Mar 31, 2008
Messages:
11,791
Really nice vid. How do you go frame by frame on that?

37. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
Even if we put constraints on the type of serve, etc etc, the fact of the matter is that the bounce puts too many factors into play. If it hit the line, then the coefficient of kinetic friction (mu) is less. If the surface is dusty, then mu is less. If the surface is clean and recently resurfaced, then it will be higher. If the ball is flat or old, that will change it's bounce characteristics. If the angle of impact is less, it will bounce less, if you're taller, it will bounce higher. Where the ball bounces in the box will of course affect things as well. Even your altitude will play a significant role in how high the bounce goes. And this is all without discussing probably the most important factor: spin.

Simply put, there is no way you can create an even remotely accurate formula for the height of the bounce at the wall to a serve's initial speed. It's a fallacy, and nobody should pay any attention to that type of thing.

Last edited: Oct 25, 2009
38. ### SourStrawsRookie

Joined:
Nov 12, 2008
Messages:
327
I say he's legit...Nice calcs

S.S.

39. ### Mike CottrillHall of Fame

Joined:
Sep 3, 2004
Messages:
1,626
Still waiting for vimeo to add this feature to their site.
However, it would work something like what can be seen here:
http://www.hi-techtennis.com/serve/martin_serve_open.php

If you create your own video, there are many different video players that can step frame by frame.

40. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
I have VLC media player, and I can view my videos frame by frame, but what software would allow me to create a video at a frame-by-frame pace? I would've added that into this video if I knew how.

41. ### Mike CottrillHall of Fame

Joined:
Sep 3, 2004
Messages:
1,626
Not sure I follow. Do you mean loaded on the internet for others to step frame by frame?

42. ### tfm1973Semi-Pro

Joined:
Jul 31, 2007
Messages:
792
very nice serve. looks legit 120mph to me too. if you want the naysayers to not pick your calculations apart - just get a radar gun.

actually even if you have a radar gun - i'm sure they would give you grief.

still doesn't change the fact that you have a nice serve.

43. ### ximianRookie

Joined:
Oct 4, 2005
Messages:
319
Yeah, a frame-by-frame video on youtube or something.

Thanks! And really, a radar gun would just pose its own set of problems. Where is the gun pointing, where is it picking up the path of the ball, and other complications. I think this method actually gives a more accurate result, even though it will be +/- 3mph or so.

44. ### Mike CottrillHall of Fame

Joined:
Sep 3, 2004
Messages:
1,626
I do not know of an internet site like YouTube or vimeo that has this feature. It would be nice if they did. I believe Jeff uses adobe flash and sites like tennis one use quick time.

45. ### NovesonHall of Fame

Joined:
Feb 27, 2006
Messages:
4,964
Location:
PDX
I've attempted to return this serve Can agree to the fact it's up there in the MPH's

46. ### ubermeyerHall of Fame

Joined:
Mar 1, 2009
Messages:
2,176
Location:
Texas
I use that for a serve sometimes, which is why i posted it. I don't remember where you said flat serve, but whatever

47. ### NovesonHall of Fame

Joined:
Feb 27, 2006
Messages:
4,964
Location:
PDX
No way you can get a lob in the service box to go over the back fence.

Last edited: Oct 25, 2009
48. ### onehandbhLegend

Joined:
Dec 19, 2005
Messages:
5,130
Nice videos, serves, and calculations!
The only other variable you are missing is drag due to spin. Probably pretty
insignificant in terms of contribution to the overall speed calculation though.

Are you a physics major or just good at math?

I miss playing indoors. Perfect conditions, no sun, no wind, better for serve &
volleying since it's harder to lob. Will have to wait until x-mas break until
I play indoor again.

49. ### ubermeyerHall of Fame

Joined:
Mar 1, 2009
Messages:
2,176
Location:
Texas
willing to place bets on that?

50. ### VermillionBanned

Joined:
Dec 30, 2007
Messages:
1,229
Looks to be in the 110-120 range for sure.