Car Crash

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Car Crash

Postby Мастер » Sun Mar 05, 2017 4:01 am

So, over at an astronomy forum, a user called BetaDust asked why in the famous E=mc2 equation, the speed of light is squared?

One of the answers given was a reasoning-by-analogy sort of response, by someone with username DaveC followed by his inmate number, who cites the formula for kinetic energy, containing velocity squared. He further explains that this is why a 40-km/h collision is much more devastating that a 20-km/h collision, because the amount of energy involved is four times as much.

It is this response which I am thinking about now, and something about it bothers me. I am possibly coming to the conclusion that it has the essence of the truth, but there are some hidden assumptions here. So I want to do some thinking out loud, and see what everyone says.

So, the basic argument is, kinetic energy is m v2/2, so if you double the velocity of the car, you quadruple its kinetic energy. So the transfer of kinetic energy is four times as much. This sounds reasonable, and it is a type of argument I have thought out (and possibly stated) myself before. Except . . .

Suppose the car is passing another car driving at 20km/h, and then crashes into the brick wall in the passing lane. Since there is brick wall in the non-passing lane, the car that was passed continues unimpeded.

Then from the perspective of an observer on the side of the road, the car that crashed went from 40km/h to 0km/h. So the kinetic energy changes from 800*m to zero.

However, from the perspective of the driver of the car that was overtaken, the crashing car was travelling at +20km/h, and then -20km/h after the collision. So the kinetic energy changes from 200*m to 200*m. Even if we allow for a directional component, we can view the kinetic energy as changing from 200*m to zero, and then 0 to 200*m, for a total change of 400*m. So half as much transfer from this perspective. Obviously the amount of damage cannot depend on the perspective of the observer, so something is missing here.

To eliminate the directional issue, let us make it a spacecraft flying through open space at 40km/h, and crashing through a wooden barrier that slows it down to 20km/h. Then the change in kinetic energy is from 800*m to 200*m, so a net change of 600*m. However, from the perspective of a spacecraft travelling 20km/h that had just been overtaken, the crashing spacecraft changes its velocity from 20km/h to 0. So the kinetic energy changes from 200*m to 0. So the change in kinetic energy is three times larger or smaller, depending on perspective. The damage done by the collision must be invariant, though. So again, something is missing from the analysis.

So the only two possibilities I can come up with here are, the idea that the damage done by the impact will be proportional to the kinetic energy transferred, is just wrong, or we need to take into account the change of kinetic energy in the object that is stricken as well.

So, it's a bit tedious, but when I take into account the change in kinetic energy in both the car and the barrier, I get that this is invariant. Note that the change in the velocity of the barrier can be calculated using conservation of linear momentum.

So the conclusion I am going to is that DaveCXXXXX's explanation may be the right idea, but it is somewhat incomplete (or perhaps just rests on an unstated assumption). Then the barrier changes in velocity from zero to <very small>, the change in kinetic energy is <very small> squared, which is <super small>. But when the barrier changes from <a lot> to <a lot>+<very small>, when we square the velocity, we get a term proportional to <a lot>*<very small>, which is <not negligible>. So when we change perspective, the amount of kinetic energy lost by the car/spacecraft changes, but the different is picked up by the barrier, so the total change is invariant.

So does that sound reasonable? The total change in kinetic energy is invariant of perspective, and this loss of energy is what goes into doing the damage. How the damage is divided up between the car/spacecraft and the barrier, this I do not know.
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Re: Car Crash

Postby MM_Dandy » Mon Mar 06, 2017 5:18 pm

From the frame of reference of the overtaken car/spaceship, does the barrier have a velocity?
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Re: Car Crash

Postby Мастер » Tue Mar 07, 2017 5:27 pm

MM_Dandy wrote:From the frame of reference of the overtaken car/spaceship, does the barrier have a velocity?


Yes.

I have it worked out quite generally now, the change in total kinetic energy before and after is invariant to reference frame, as long as linear momentum is conserved. That seems good, otherwise, the amount of energy released in the collision would depend on perspective, which doesn't feel right.

However, the point I'm wondering about now is - two objects smack into each other. A certain amount of energy is released in the collision. How is the energy unleashed divided between the two objects? I don't really know how that works.
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Re: Car Crash

Postby tubeswell » Tue Mar 07, 2017 5:32 pm

Мастер wrote:
MM_Dandy wrote:From the frame of reference of the overtaken car/spaceship, does the barrier have a velocity?


Yes.

I have it worked out quite generally now, the change in total kinetic energy before and after is invariant to reference frame, as long as linear momentum is conserved. That seems good, otherwise, the amount of energy released in the collision would depend on perspective, which doesn't feel right.

However, the point I'm wondering about now is - two objects smack into each other. A certain amount of energy is released in the collision. How is the energy unleashed divided between the two objects? I don't really know how that works.


Maybe the energy doesn't transfer to each other object, but maybe it zings away in some sub-atomic energy transfer. Maybe it goes into another dimension.
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Re: Car Crash

Postby MM_Dandy » Tue Mar 07, 2017 5:43 pm

A lot of energy is transferred during the deformation of the colliding objects. Otherwise, I suppose that there's heat generated due to friction, the expulsion and deflection of debris, the sound of all the bits and pieces crunching, etc.
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Re: Car Crash

Postby Arneb » Tue Mar 07, 2017 6:39 pm

I think this might be what Mactep is looking for.

I admit I didn't quite see the problem with the reference frame (of course, only if it's non-relativistic, and velocities simply add up), as in both frames, the passing car lost 1/2*its mass * 40^2 kg*m^2/s^2 in kinetic energy, going from 40 to zero in one, and from 20 to -20 km/h in the other frame. What was I missing?
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Re: Car Crash

Postby Мастер » Tue Mar 07, 2017 11:28 pm

Arneb wrote:I think this might be what Mactep is looking for.


Will have a look.

Arneb wrote:I admit I didn't quite see the problem with the reference frame (of course, only if it's non-relativistic, and velocities simply add up),


Yes, non-relativistic.

Arneb wrote:as in both frames, the passing car lost 1/2*its mass * 40^2 kg*m^2/s^2 in kinetic energy, going from 40 to zero in one, and from 20 to -20 km/h in the other frame. What was I missing?


In both frames, it loses m*v12/2-m*v02/2. When the car goes from 40 to 0, this is 800*m; when it goes from +20 to -20, the change is 0. In some frames, the car gains rather than loses kinetic energy. But the resolution to that is to take into account the change in kinetic energy of the barrier as well. The total change in kinetic energy, of the car plus the barrier, is invariant provided linear momentum is conserved.
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Re: Car Crash

Postby Мастер » Wed Mar 08, 2017 5:30 am

MM_Dandy wrote:A lot of energy is transferred during the deformation of the colliding objects. Otherwise, I suppose that there's heat generated due to friction, the expulsion and deflection of debris, the sound of all the bits and pieces crunching, etc.


Yes, which causes me to think, the answer might be really complicated, depending on the shapes of the objects, their compositions, etc.
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Re: Car Crash

Postby Arneb » Wed Mar 08, 2017 2:52 pm

Мастер wrote:When the car goes from 40 to 0, this is 800*m; when it goes from +20 to -20, the change is 0.


But is it? I seem to remember that velocity and kinetic energy being voectors, the direction does matter.

ETA: A delta v of 40 km/h will take the same amount of fuel (energy) no matter if it's from 40 to zero, zero to 40 , 20 to -20, or -20 to 20. Or 299,960 to 300,000, non-relativistically. Won't it?
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Re: Car Crash

Postby Мастер » Thu Mar 09, 2017 12:35 am

Arneb wrote:
Мастер wrote:When the car goes from 40 to 0, this is 800*m; when it goes from +20 to -20, the change is 0.


But is it? I seem to remember that velocity and kinetic energy being voectors, the direction does matter.


Velocity is a vector. Kinetic energy is a scalar, it is the inner or dot product of the velocity vector, times mass over two.

Let us suppose we are moving strictly in the x-dimension, and velocity changes from (40,0,0) to (0,0,0). The kinetic energy initially is m*(402+02+02)/2=800*m. After, it is m*(02+02+02)/2=0.

If instead the velocity changes from (+20,0,0) to (-20,0,0), the kinetic energy initially is m*(202+02+02)/2=200*m. After, it is m*((-20)2+02+02)/2=200*m.

Arneb wrote:ETA: A delta v of 40 km/h will take the same amount of fuel (energy) no matter if it's from 40 to zero, zero to 40 , 20 to -20, or -20 to 20. Or 299,960 to 300,000, non-relativistically. Won't it?


I think for spacecraft, fuel and delta-v are equivalent. I do not know whether that holds on the autobahn. However, it seems like if the different conditions are just due to different frames of reference, it doesn't make a whole lot of sense for one observer to note that the fuel tank is full, while the other says it is empty. This is one of the things that seemed initially puzzling to me. But I think is this case, the resolution to the puzzle is the same as my example with the car crashing into a barrier. The change of kinetic energy of an individual object is not invariance to frame, but the total change of kinetic energy of all objects is invariant, as long as linear momentum is conserved. When you are negotiating a hard turn on the autobahn, you are changing the rotational velocity of the earth as well. So the total change in kinetic energy (car+earth) should be invariant to the reference frame.
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