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# Roger's Equations

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### Impulse - Why Seatbelts Work

Posted June 06, 2007 2:47 PM by Roger Pink
User-tagged by 1 user

Impulse

Isaac Newton's second law of motion is

F = ma

where F is the force, m is the mass, and a is the acceleration of an object. Acceleration is just the change in velocity over time.

a = Δv/Δt

where Δv is the change in velocity and Δt is the change in time. So we can rewrite the second law as:

F = m (Δv/Δt)

Notice the larger Δt is for a fixed m and Δv, the smaller the force on the object is.

If you multiply both sides of the above equation by Δt you get the following equation:

J = FΔt = mΔv

where J is the physical quantity called Impulse. For those of you wondering why its J and not I, yeah, me too.

Now remember that

P = mv

so for a fixed mass

mΔv = ΔP = J

So impulse is really the change in momentum.

Surviving Impact

"It's not the fall that kills you, it's the sudden stop at the end."

Seatbelts and airbags works because they increase the time it takes for our bodies to slow down in a car accident, thus reducing the maximum force exerted on the body.

http://auto.howstuffworks.com/seatbelt.htm

This is the same reason why some of the most lethal accidents are sudden and relatively mild looking. If a car going 200 mph flips over and rolls three times before finally coming to a stop, the maximum force on it isn't nearly as large as a car that slams into a wall at 200 mph and suddenly stops.

Skydivers are taught to tumble as they land. This is because tumbling spreads out the force of impact over time.

Water is great at slowing people down slowly. That's why we can dive into a deep pool from as high as 15 ft without any fear.

Much of the martial arts depends on the idea of minimizing the time used to deliver maximum force.

http://www.fas.harvard.edu/~scdiroff/lds/NewtonianMechanics/KarateBlow/KarateBlow.html

As far as safety equipment goes, the idea is to maximize time to minimize the force experienced by the body. Air bags and seatbelts take longer to slow you down then say a windshield or a dashboard, so the injuries tend to be bruises rather than broken bones.

Till next time.

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Anonymous Poster
#1

### Re: Impulse - Why Seatbelts Work

06/09/2007 11:08 AM

Tut tut - if the same force lasts for longer it is more likely to kill you, not less. What you presumably mean is that it is not simply the amount of KINETIC ENERGY in the impact that kills you, but the time over which it is implied. And the reason that the longer times are less damaging is that the energy is then absorbed by something other than your body.

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#2

### Re: Impulse - Why Seatbelts Work

06/09/2007 12:44 PM

You were correct that my last sentence was technically incorrect, I have fixed it. However, your suggestion for a replacement is incorrect as well. You suggest that it is the time over which kinetic energy is applied. This is not correct. For example:

Consider two cars, one in front of the other, on a straight road. The first car, which starts out 100 m ahead is moving at 20 m/s. The second car is moving 20.1 m/s. The mass of both cars are 1000 kg. Since the car in front is moving slower than the car behind it (by .1 m/s) it's clear that there will be a collision between the two cars eventually. The impact of that collision will be the same as if the front car was stationary and the trailing car was traveling at .1 m/s.

Clearly the total kinetic energy of the first example (20 m/s) and second example (.1 m/s) above are very different, so when you said it's not "simply the amount of KINETIC ENERGY in the impact that kills you, but the time over which it is implied" is incorrect. In the example above it's clear that the collisions are equivalent despite the fact that one scenario (car at 20.1 m/s) has a much higher kinetic energy than the other scenario (car at .1 m/s).

It is the change in momentum that matters. The longer it takes for the change in momentum to occur, the smaller the force felt by the mass experiencing the change in momentum.

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#3

### Re: Impulse - Why Seatbelts Work

06/11/2007 7:13 AM

Sloppy of me - should have written change in kinetic energy (Newtonian Relativity?)

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#4

### Re: Impulse - Why Seatbelts Work

06/11/2007 10:02 AM

I'm still not seeing how a "change in kinetic energy" applies here. Consider the following example:

Two rocks with equal mass (M) approach each other frictionlessly. The rocks are 100 m apart and traveling in toward each other at 10 m/s. The mass M is small so we can neglect gravity. For this system the Kinetic Energy is:

K.E. = 1/2 MV12 + 1/2 MV22 = 1/2 M(10 m/s)2 + 1/2 M(-10 m/s)2 = 100 kg·m2/s2

The rocks collide elastically and travel in the opposite directions. The total kinetic energy for this new system is:

K.E. = 1/2 MV12 + 1/2 MV22 = 1/2 M(-10 m/s)2 + 1/2 M(10 m/s)2 = 100 kg·m2/s2

Notice that although there was a fairly violent collision, there was no change in kinetic energy.

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#5

### Re: Impulse - Why Seatbelts Work

06/11/2007 10:25 AM

And no damage to either of the rocks... but that is merely being irrelevantly pedantic.

But you do correctly highlight an issue - that you can't simply say either impulse or change in kinetic energy without massive qualification. The problem with trying to use impulse is that the level of damage varies far more than linearly with the size of the impulse. Even assigning a threshold force (or pressure - depending on the dynamics) doesn't work. If you perform all the calculations with either the initial velocity or the final velocity of the damaged object as the reference, energy fits better than any other simple measure. You could think of this in terms of the peak force x the distance over which it operates...

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#6

### Re: Impulse - Why Seatbelts Work

06/11/2007 10:57 AM

I think we're getting off track here. My entry here was as discussion of impulse with some examples such as a description of why seatbelts and airbags prevent injuries. The point of the entry was that it isn't enough to just talk about a change in velocity, one has to talk about how long that change in velocity lasted in order to get an accurate measure of the force experienced by the accelerating object. For example, it the reason why rubber bullets are nonlethal. They fly just as fast as regular bullets, but they don't break the skin because when they make contact they take longer to slow down and thus apply less force.

You wrote "If you perform all the calculations with either the initial velocity or the final velocity of the damaged object as the reference, energy fits better than any other simple measure."

That's not correct. What we are talking about here is force, and change in momentum is how you talk about force, not change in energy.

After all:

F=dp/dt

Think of it the opposite way. Lets say I've got a 10 kg block and I want to accelerate it to 10 m/s. If I have 10 seconds to accelerate it I will need much more force than if I had 1,000,000 seconds to accelerate it. At the beginning (when they are stationary) the kinetic energy in both scenarios is zero. After acceleration they both have velocity of 10 m/s and thus the same kinetic energy of 500 kg·m2/s2 The difference in kinetic energy of both systems is 500 kg·m2/s2 , both are equal. Yet the force experienced in the first scenario is much, much larger than that of the second scenario.

I hope you see that it's not the change of energy of a collision (after all, there can be cataclysmic collisions with no net energy change) but rather how long a change in momentum takes that tells you how much force will be experienced by the object doing the accelerating.

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#7

### Re: Impulse - Why Seatbelts Work

06/11/2007 11:14 AM

I couldn't resist weighing in here -

I agree with you both that you need to be careful about definitions. However, once you exceed the damage threshold, excess energy over threshold is a much better fit to observed damage than anything excess momentum (impulse). SFIK, (force - threshold)*distance gives a much better fit to actual damage than (force - threshold)*time.

The title was "why seatbelts work" - and the reason is that the body's kinetic energy is absorbed over a long enough distance that the critical forces are not exceeded. The design constraints for crumple zones to assist the seat belts are force and distance.

Fyz

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#8

### Re: Impulse - Why Seatbelts Work

06/11/2007 11:27 AM

I would appreciate it if either of you would provide an equation that shows how a change in kinetic energy relates to the force experienced by an object. That may go a long way in helping me understand what you both are talking about. I'm very concerned that the physics is being misrepresented in this discussion and that's when I turn to math. For instance:

F = m (Δv/Δt) is straight forward. If you fix the change in velocity and the mass, the force experienced depends on the time the change in velocity took place.

also since mΔv=Δp

F= Δp/Δt is another way of describing the situation.

Please use an equation like I just did to explain how energy, kinetic energy, change in kinetic energy, etc. is the best way to describe what's going on.

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#9

### Re: Impulse - Why Seatbelts Work

06/11/2007 12:21 PM

Assume that we are designing a crumple zone to limit the peak deceleration starting from some nominal velocity (maximum protected velocity). The time depends on the peak-velocity, and the length of the crumple zone is average-velocity x time - or Kinetic-energy/mass_of_vehicle.

Mathematically the same can apply to damage. We need a case we can define clearly. As a first example, we can consider an object of fixed frontal area and high mass entering an object that has a fixed resistance to motion (this is not typical of a bullet entering a human - follow the links in Wikipedia physics_of_firearms for basic information on that). The distance of penetration will again be proportional to the kinetic energy, not the momentum. This is more-or less equivalent to a steering wheel crushing against your chest - though that particular case is more peaky, because more critical regions are easlily damaged once the ribs are broken.

Obviously, the second case is not the whole story - you'd need to look at accident analyses in detail to get a proper picture - but it's a start.

Regards

Fyz

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#10

### Re: Impulse - Why Seatbelts Work

06/11/2007 12:59 PM

That still isn't an equation relating Force and Kinetic Energy. Please provide an equation so we have something specific we can discuss.

Did you have any comments on the equations I provided? Did they look correct to you? Doesn't that seem like a simpler way of explaining why an airbag works than some indirect measure of energy?

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#11

### Re: Impulse - Why Seatbelts Work

06/11/2007 1:09 PM

Sorry - did you mean this?

E = F * L

If you want it to be kinetic energy, you have to define your inertial frame of reference. Suitable frames would use either the velocity of the object_to_be_protected at the point where the force reaches the damaging threshold, or the velocity at the point where the force again falls below the damaging threshold. The difference in change of KE is just the sign.

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#12

### Re: Impulse - Why Seatbelts Work

06/11/2007 1:16 PM

So you're saying the equation

F = E / L should be used to describe why an airbag prevents injury.

What Energy would it be and what would length would be measured? Can you give an example with numbers so I can understand what you're saying better.

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#13

### Re: Impulse - Why Seatbelts Work

06/11/2007 1:36 PM

The protective length is the length of the car that is available to crumple during collision. You can include the distance you move into the seat-belt and/or air-bag within that. In practice, the kinetic energy of interest is the one that causes the damage - that could be just your head, or your torso, depending on which part of the body you are considering.

Example (probably not representative)
42-mph, 10gn peak acceleration -> 6-feet crumple zone etc required.

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#14

### Re: Impulse - Why Seatbelts Work

06/11/2007 3:06 PM

You wrote: "In practice, the kinetic energy of interest is the one that causes the damage - that could be just your head, or your torso, depending on which part of the body you are considering."

So are you suggesting that if there is no permanent damage after a collision, there is no force exerted during a collision since E=0 in the equation F=E/L = 0. What about billiard balls colliding? The change in direction (thus change in momentum) suggests a force in the collision, yet there is no damage to the billiard balls after the collision.

"Example (probably not representative)
42-mph, 10gn peak acceleration -> 6-feet crumple zone etc required."

Although you qualified it by saying it's not representative, I'll point out your example assumes too much to make your point. I basically would have to take your word for it as it has many variables built in such as how the car crumples, the material it's made from, the hardness of the material it hits, etc.

I'm buying the length part of your argument, I'm just not clear on what the energy represents in the collision. Conservation of energy would seem to get in the way in any collision where there isn't permanent damage.

Perhaps energy of deformation during the collision? I could see that working, but is that really an easier approach than just measuring the change in momentum?

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#15

### Re: Impulse - Why Seatbelts Work

06/11/2007 5:39 PM

I obviously failed to communicate my points. I'll try harder.

First: the crumple zone is designed on the basis of the expected loaded weight of the car, so that the deceleration does not exceed some nominal value within the design rating. The calculations tell you what the dimensions of the crumple zone must be - the point of the artificial numbers was to show how the requirement is related to kinetic energy, rather than to momentum (or impulse). You need to know about the materials and the weight of the car to design it - but not in order to know how to specify it.
The elastic portion of deformation is included in the calculations of the crumple zone - so it does not represent the permanent deformation of the vehicle.

Second, the pre-damage level of impact on the occupant:- I mentioned in a previous post earlier, but didn't subtract explicitly it from the totals for two reasons - first, the acceleration during crumpling takes up slightly more than the pre-damage level - you should expect bruising at the point where the vehicle crumples, otherwise you are unnecessarily restricting the maximum level against which you protect against more severe injury. So any excess beyond this that affects the occupants will damage them - the difference between permanent and temporary injury is in the level of repair you can achieve. The other reason was that I was working solely on the reason that the factor was kinetic energy - the basis for the factors governing excess would be no different than if the critical feature was momentum - and you had already covered this (presumably to your satisfaction).

Clearly, even billiard balls will be damaged if you change their kinetic energy by a large enough amount. (Actually, based on what I discovered about steel balls when struggling with the detail* of the bouncing ball problem, I suspect that the surface is rearranged under normal use, but not deeply enough to cause problems)

Fyz

*What happens if you can arrange that the balls are in contact at the moment of collision with the ground. (The information on damage wasn't relevant to the problem, but I saw it en-passant, as it were)

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#16

### Re: Impulse - Why Seatbelts Work

06/12/2007 9:10 AM

You Wrote: "I obviously failed to communicate my points. I'll try harder."

No, you made your points clear the first time. You just don't seem to have any interest in what I'm trying to say to you.

Maybe its my fault, maybe I'm not communicating my points clearly. Let me list them.

1. Damage to a person in an accident is due to force(s) experienced by that person.

2. Impulse (FΔt) is describes the average force experienced during a collision.

3. The impulse of the force acting on an object equals the change in momentum of that object. College Physics, Serway/Faughn (Sixth Edition)

From those facts 1,2,3 above; I draw the following picture

1. Airbags / Seatbelts reduce the force felt by the driver during an accident.

2. Since the force felt by the driver during an accident has been reduced and the change in momentum is the same as without an airbag, given the equation:

FΔt = Δp

Since F is smaller and Δp is fixed, then Δt must be larger

3. Thus airbags / seatbelts protect us by having the change in momentum of a collision last longer (Δt is larger).

Energy

Energy is indirectly related here (as it is in all physics). If there were no change in momentum, there is no collision. Crumple zones work because they lengthen the time it takes for the car to slow down, to change momentum. Sure you could say that it's absorbing energy, any example where there is damage there has been absorbed energy (obviously). What about collisions where there isn't damage? I'm sure you can talk about deformation energy if you wanted to.

Look, energy is involved in all physical processes. Suggesting "energy is absorbed" doesn't mean you know a damn thing about the physics involved. Look, I'll show you:

"quarks bind together to form a proton because they save energy by doing so"

See how useless a statement that is? It's true, no doubt about it, but it isn't physics. That's what your doing here.

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#17

### Re: Impulse - Why Seatbelts Work

06/12/2007 1:36 PM

No, I have failed to convey my case.

First - our area of agreement: we agree that the level of force or pressure determines the onset of damage.

Now my case: it is that the magnitude of the impulse is irrelevant to whether damage is done. For that, you need to look at the kinetic energy that is lost and where it is lost.

Consider a passenger in a car that is slowed from 120-mph to 0-mph. The momentum change (or impulse) that is conveyed to the passenger is equal to his mass times the change in velocity; this is irrespective of how long it takes for the passenger to be decelerated. If it happens quickly as in a crash into a solid object, the passenger will likely be killed, if it happens gradually as by braking, the passenger will suffer no damage at all. But, in both cases, the same impulse has been conveyed to the passenger.

I did not previously make my case in these terms because that was done more than adequately in the Wikipedia article on firearms that I referenced. There they consider the impulse that is delivered to the person who fires a round, and to the person hit by a rubber bullet or a conventional bullet. In all three cases the impulse is identical, but the energy transferred to the recipient is very different. The firer experiences a jolt, the recipient of a rubber bullet has the wind knocked out of him, and the recipient of a lead bullet will suffer significant damage (depending on where the bullet hits).

In all cases, we have identical levels of impulse imparted to the recipient - and it has no direct bearing on the level of damage. In fact, there is no practical minimum level of impulse below which damage does not occur, and no practical maximum level of impulse above which it always occurs.

What does the damage is energy conveyed to the recipient, which in these cases is kinetic energy converted to mechanical action in the recipient.

I also gave an example of where kinetic energy can be used directly to calculate the dimensions needed to bring the force down to an acceptable level. I know of no equivalently simple algorithms that involve impulse. If you can produce one, I will be content to accept what I do not as yet - that impulse can have some validity in this context.

Regards

Fyz

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#18

### Re: Impulse - Why Seatbelts Work

06/12/2007 2:03 PM

Fyz,

I know you're a smart guy from past discussions, but if you don't listen to what I'm saying rather than reacting to what I'm saying then we are going to get nowhere.

You wrote: "The momentum change (or impulse) that is conveyed to the passenger is equal to his mass times the change in velocity; this is irrespective of how long it takes for the passenger to be decelerated (Roger - True). If it happens quickly as in a crash into a solid object, the passenger will likely be killed, if it happens gradually as by braking, the passenger will suffer no damage at all (Roger - True). But, in both cases, the same impulse has been conveyed to the passenger (Roger -True)."

Impulse=FΔt=constant

If FΔt = constant, the larger Δt is, the smaller F is. That's it. That's all there is to it. I don't know how to make it easier than that.

You Wrote: "I also gave an example of where kinetic energy can be used directly to calculate the dimensions needed to bring the force down to an acceptable level. I know of no equivalently simple algorithms that involve impulse."

I've written the equation a ton of times. See above.

Look Fyz, if after reading this you still disagree with me, forget about it, we are getting nowhere. I don't know how to explain it any clearer and I don't want to be rude to you but I'm out of patience.

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#19

### Re: Impulse - Why Seatbelts Work

06/12/2007 3:17 PM

It is sad when you have to threaten with lack of manners.

No, I hear you, and I think I understand what you have said - I am responding to what I perceive to be missing. I.e. that I can't see where you have justified for using momentum in preference to other measures.

You are saying (and I agree) that force is an important parameter and F=d/dt(momentum) - that is force (or divide by mass and/or area and use acceleration etc.). Obviously, this equation does not show where using momentum (or velocity) in addition to force provides any additional information. In order to justify the use of momentum as a prime measure (as opposed to say energy), you would have to show how it is a more suitable predictor for protection or for damage.

That is why I gave one specific case of how energy helps with calculating the crumple zone (protection), and cited the Wikipedia article on ballistics - which links to all sorts of additional stuff that it would be too time-consuming to list here.

I've just realised that you might be thinking that I have failed to justify KE because I haven't listed the equivalent to your momentum equation - so here it is:
F=d/dL(kinetic_energy)

Regards

Fyz

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#20

### Re: Impulse - Why Seatbelts Work

08/13/2008 4:28 AM

Nice post. But design of airbags, seatbelts, elevators are not based solely on the acceleration (second derivative of length with respect to time), but rather on the jerk which is the (third derivative of length with repect to time) or the derivative of acceleration.

What's more the complete mathematical modeling of impact is complex using several non linear transcendental differential equations, where objects are connected by springs and dampers.

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