# I Tennis ball hits head

1. Feb 28, 2016

### paget

I have a question concerning GeForces. imagine a tennis ball that weighs 57 Gramm Hits a human head that weighs 4 Kilogramm on the forehead, at a speed of 120 kilometers per hour. the head moves back 15 centimeters until it stops because of the resistance of the spine. how much g-force did the impact create?

2. Feb 28, 2016

### BvU

Hello Paget,

Do you have this detailed question or is it your teacher, your textbook or some other source of homework ?

In that case, filling in the template is only don for one third and you still have some work to do:

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

3. Feb 28, 2016

### paget

This happened to me in real life a while ago. I finished school a long time ago. is it not possible to calculate this with data I provided?

4. Feb 28, 2016

### jbriggs444

If you are after the acceleration produced by the tennis ball on the head, one could make some reasonable estimates based on conservation of momentum and the size of the tennis ball. Start by converting everything to meters, kilograms and seconds. So that's $120 \frac{km}{hr} \cdot \frac{1000\ meters}{1\ km} \cdot \frac{1\ hr}{3600\ sec} = 33 \frac{meters}{sec}$

What is the change in momentum of the tennis ball as it comes to a stop upon hitting the head? [Just talking impact here and not rebound]

Given Newton's third law, what is the change in momentum of the head as a result of the impact?

Given the mass of the head, what velocity would that be?

A big unknown is how much the tennis ball compresses during the impact. The size of the tennis ball puts some bounds on that. It cannot compress by more than its own size. Assume that the tennis ball and skin jointly compress by 2 centimeters (0.02 meters) during the collision. At an average speed that is half of 33 meters/sec, how long will the impact take?

You now have a change in velocity and a reasonable guess at the time interval over which it changed. That is enough to compute an average acceleration. You do not need to know the 15 cm that the head moved as a result of the impact.

5. Feb 29, 2016

### paget

Thank you briggs.

I knew that it will be difficult to get a close result But I didn't know that the calculation is that difficult.

If I understood you right the last question is how long the impact takes. I would say a reasonable guess is 0.5s

6. Feb 29, 2016

### jbriggs444

What is the average velocity of the tennis ball during the impact? (just impact, ignore rebound). How far would the tennis ball have moved in 0.5 seconds? How much distance was there available for it to move?

7. Mar 1, 2016

### paget

the first question it's hard to estimate, what would you say? the tennis ball moved 5 mm until it stopped.

8. Mar 1, 2016

### sophiecentaur

I would suspect that the movement / distortion was more than 5mm - if it hurt!
We have a regular input of questions about impacts / collisions / injuries, on PF. It is a shame that there is almost never a satisfactory answer available. This is because there is never sufficient data. If you had a high speed film of the event then you could estimate the distances travelled during the impact and the distortion of the ball but, off the top of your head (pun intended) there is no way of knowing any accurate quantities. You could get some way along the path for an answer, though.
When a tennis ball hits a surface, it distorts a lot and you could, perhaps assume a certain amount (say it flattens to about 50% of its diameter - a complete guess). You could then measure the force needed to distort it statically, using a bathroom scales. That could give you a ball park figure (OMG, there's no limit to ball-oriented puns here) for the sort of force your head was subjected to. On the net there are images of ball collisions and you could browse through them to get a better idea of actual distortion.
Once you have a reasonable estimate of force, you can get some idea of the time of impact. What happens to your head is a more complicated matter but you could do some basic collision sums*, treating the head as an rigid mass and finding the mass value from a further internet search.
Apologies but I may have unleashed a set of typical PF puns, based on the word ball.
*Look on the Hyperphysics site for calculations of colliding bodies.

9. Mar 2, 2016

### paget

sophie, I understand that the physics behind the impact are very complicated, but this calculation should just demonstrate a reasonable result, even it if varies from the actual event.

I would say that the tennis ball moved 8 millimeters at maximum, and yes it did hurt.

10. Mar 2, 2016

### sophiecentaur

OK - 8mm. You could find the force needed to distort the ball by 8mm and that would give you something like an answer. BTW, how did you arrive at the 8mm figure? I have seen high speed pictures of bouncing balls and they distort by amazing amounts. Those would probably take your head off, though!
This question has a lot of facets which could be answered roughly using kitchen table experiments.
One relevant quantity in these events is change of Momentum which is Force X time, or Impulse. It may be possible to experiment with a tennis ball to find the time of impact by treating the system as a simple harmonic oscillator (gross oversimplification but a start. Treating it as a simple spring, you could find the spring constant, k by plotting a stress / strain graph. You could then measure its mass to get the period of oscillation.
T = 2π√(m/k)
The impulse would be (very approx) T times Force/2, which would be transferred to your head on impact.
I would imagine that your estimated 15cm figure for head recoil is very likely due to muscular action, bearing in mind the vast difference between the masses and the probable time of actual ball-head contact. You could experiment here, as well, with a suitable representative target (turnip, hanging on a string, for instance) and a tennis ball, to see how far / fast the target is moved. Pendulum physics at work here and you could use the displacement to find the recoil speed. Hyperphysics has all the necessary equations.

11. Mar 4, 2016

### paget

I appreciate all your help but you start to make things way too complicated for me now. I was never good at physics and I can't follow you anymore, sorry.

reasonable estimations are absolutely sufficient for my caveat, like jbriggs was doing in the previous posts.

otherwise this calculation would get to scientific and I don't understand it anymore. so the last estimation we need is the average velocity during the impact. let's take 100 kilometers per hour

Last edited: Mar 4, 2016
12. Mar 4, 2016

### sophiecentaur

Not to worry Physics gets very meaty, very quickly but PF really doesn't encourage spoon feeding and people are expected to make some contribution to the effort of getting to an answer. At least, on PF, you have a good chance of getting a reliable answer; unlike on some other sites.
One thing you, personally, could do would be to push a tennis ball against some scales and to see what force is needed for 8mm of distortion.

A rough value would be 50kph because it starts at 100 and ends at near 0 and a first approach would assume constant rate of change. But the point I have been trying to make is that the time and / or distance involved in slowing the ball down are not knowable without further measurement or guessing.
If your head were free to move and the two are stuck together afterwards, the Momentum Equation would apply:
Initial Momentum = total final momentum
0.057 X 33.33 = 4.057 X final speed
(33.33m/s is 120kph)
so final speed = 0.47m/s (1.7kph)
If the ball bounces off (nearly elastic collision) the sum is harder but the head's speed could be up to twice the above.
These sums ignore how long the interaction lasts and say nothing about the forces involved or the accelerations, which is what you wanted. But measuring the squashing of the ball would tell you the force and the head's acceleration (turnip model) would be roughly this force (in N) / head mass (4).

13. Mar 7, 2016

### paget

There was a misunderstanding I meant that the skin deformed 5 and 8 millimeters ,not the ball. for the deformation of the ball and the skin we estimated 2cm (1.5 ball, 0.5 skin) I measured with a scale that the deformation of the ball of 1 centimeters requires 40 kilogram, this is as hard as I can press.

14. Mar 7, 2016

### sophiecentaur

The human body is a very hard structure to define and to experiment on. A tennis ball is a more reliable test piece; it doesn't have the wide variety of squishy and hard bits that the body has, inside and out.
The 40kg (400Newtons would be the actual Force, involved) pretty much gives you one of your answers.
Just looking at your original scenario again, that's a pretty horrendous trauma on the victim's head. An county class tennis player serving full on at your head. Concussion would be the least of your worries, I reckon.
The kinetic energy in such a projectile is mv2/2
(v would be 33.33m/s)
= 0.057X 33.3332/2
Which is about twice the maximum legal energy of an air rifle in UK. And a max power air rifle is a very underrated weapon.

15. Mar 9, 2016

### paget

So we have now 400 Newton and 32 joule, but still no G-Force?

16. Mar 9, 2016

### A.T.

17. Mar 9, 2016

### sophiecentaur

The term "G-Force" in not used in Physics, although we sort of know what it refers to. The Acceleration that's caused to the target head is, as was pointed out at the beginning, is not really knowable but as F = ma, a ball park (peak) figure would be about 10g, as A.T. suggests. If that acceleration was applied for a long time - such as in a space launch- or repeated as a vibration it would be 'noticeable'!
But the head / brain is not a simple, rigid structure and the acceleration of the skull would be more than the acceleration of the brain, in its cushioned liquid support so don't put too much significance to the bald 10g figure. But it would depend on what you actually want to get out of this exercise. (Hopefully nothing to do with an actual sustained injury.)
There are a load of hits from a google search on vibration or shock testing of equipment. 10g is quite extreme for electronic circuits when applied in the form of a constant vibration. People can get concussion from tennis ball impacts, so it's clearly to be avoided, if possible. I had concussion after being hit with a badminton racket - which is not a heavy weapon and there was not much 'wellie' behind the blow. Also you can find lots of hits about brain damage and impact.

18. Mar 9, 2016

### A.T.

http://www.ncbi.nlm.nih.gov/pubmed/12900697
At 9 m.s(-1), peak linear acceleration of the head was 158 +/- 19 m.s(-2) (mean +/- standard deviation) and peak angular acceleration was 1302 +/- 324 rad.s(-2); at 12 m.s(-1), the values were 199 +/- 27 m.s-2 and 1457 +/- 297 rad.s-2, respectively.

That's more than 20g. But I don't mean to suggest that it's healthy.

19. Mar 9, 2016

### sophiecentaur

I would be interested to know the context of the OP's question.