# How Does Newton's Laws of Motion Explain a Multi-Car Rear-End Collision?

• ng19delta
In summary, Scott's girlfriend was in a motor vehicle accident last week. She was just about stopped in stop and go traffic, and was impacted from behind, and driven into the car in front of hers, and the two moved into the car ahead of that, for a total of four cars involved.Assuming the following, the order of impacts as I believe they happened are these: Car 4 and car 3 were fully stopped, as traffic forward of them had stopped. Car 2 had come to almost a complete stop, and car 1 impacted car 2 at an unknown speed. Car 2 was then struck a second time by car 1, a result of car 2's transfer of energy into car 3, (which caused car 2 to
ng19delta
This appears to me to be a simple problem, but I am not a very big physics person- Though I do know that is the key to the problem... Short story is, my girlfriend was in a motor vehicle accident last week. She was just about stopped in stop and go traffic, and was impacted from behind, and driven into the car in front of hers, and the two moved into the car ahead of that, for a total of four cars involved.

Here are the variables as I understand them:
Weather- clear

Assumptions:
All drivers, except car 1, applying brakes.

Weights:
Car 1(car at rear struck car 2) 2630lbs curb weight. We shall assume 2800 to be the total weight
Car 2(mine) 3344lbs curb weight, we shall assume 3500lbs
Car 3(forward of mine) 3329lbs, we shall assume 3500lbs
Car 4(forward most vehicle) 2780lbs, we shall assume 2900lbs

Positions:
Car 4, at front of line, stopped
Car 3, behind car 4, stopped, we shall assume typical spacing of about 5 feet
Car 2, behind car three, almost completely stopped, we shall assume spacing of about 7 feet, diminishing
Car 1, behind car 2, moving: we need to solve for its approximate speed & force.

The order of impacts as I believe they happened are these:
Car 4 and car 3 were fully stopped, as traffic forward of them had stopped. Car 2 had come to almost a complete stop, and car 1 impacted car 2 at an unknown speed.

I believe car 2 was impacted in accordance with Newtons Laws of Motion, and was propelled forward,impacting and transferring the energy into car 3 which was propelled forward away from car 2.

Car 2 was then struck a second time by car 1, a result of car 2's transfer of energy into car 3, (which caused car 2 to slow down) while car 1 remained in motion after the initial impact, which reduced its speed, but still had sufficient momentum to strike car 2 a second time.

Car 2 was driven a second time into car 3(by the second impact from car 1), which continued forward into car 4, and all four cars came to rest at this point.

What I would like to know is:
1. Does this theory- that the rearmost impacting car(1) caused the car it hit(2) to be propelled forward striking the car ahead of that(3) & then the initially struck car (2) stopped when its energy transferred into the one it hit(3), was then impacted a second time by the rearmost car(1), driving it into the back of the car(3) ahead a second time, and the group into the front car(4)- sound like a reasonably probable chain of events;

2. WHAT would the speed of the rearmost car (1) potentially have been to make such a series of events likely?

3. What was the FORCE of car 1 into car 2(first time); car 2 into car 3 (first time); car 1 into cars 2, 3, & 4?Sorry for a LONG question, but I was watching "Big Bang Theory" while I was formulating this. I simply want to find an answer, as we are having to go to court because the driver of car 3 claims my girlfriend's car(car two) struck them TWICE, so she has received a summons for "Careless Driving" !(And SHE was the FIRST ONE HIT!). I want to have a pretty good idea of what the exact physics were involved, and I think someone here may be able to help... :)

Thanks!
ScottThey say ∏ r square. To me, ∏ r ∇∇∇∇∇∇ all neatly arranged in a circle, with the tips together... Until I eat one... After a while, ∏ r GONE.

Hi ng19delta! http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif

I'd say you have pictured the possibilities fairly clearly. With car 3 being shunted twice, if car 2 was just the meat in the sandwich then it's likely car 2 was struck twice in the rear, too. So does driver 2 recall receiving two hits from behind? Maybe there is evidence of "dents within dents" on the rear of car 2 or the front of car 1, if you can access it to take photos.

Last edited by a moderator:
We have had quite a number of similar questions over the years. The only correct answer is that you have to hire a professional accident reconstructionist. Only his/her testimony can be of any use in a court case. Anything calculated here, or by you, would be considered either hearsay or speculation by a judge.

Thanks for the replies! I am pretty much just trying to figure the numbers, not use answers here for anything but guidance...

At this point, I have the simple equation f=Ma, which, for the rearmost car is using the variables M=2630lbs, a= 73fps(est. 50MPH) and F=191,990lbs

I am working out the rest...

Thanks!
Scott

I simply want to find an answer, as we are having to go to court because the driver of car 3 claims my girlfriend's car(car two) struck them TWICE, so she has received a summons for "Careless Driving" !
I think it is clear that the driver of car 1 caused an accident by crashing into car 2. It is highly unlikely that two drivers cause independent accidents at the same time (compared to situations with just one), which "careless driving" of car 2 would suggest.

With billard balls, the scenario described in the first post is possible. With cars, see Danger's post.

ng19delta said:
Thanks for the replies! I am pretty much just trying to figure the numbers, not use answers here for anything but guidance...

At this point, I have the simple equation f=Ma, which, for the rearmost car is using the variables M=2630lbs, a= 73fps(est. 50MPH) and F=191,990lbs

I am working out the rest...

Thanks!
Scott

Sorry, but it really isn't as simple as you'd like it to be. As stated above, your best bet of getting an accurate answer would be an accident reconstructionist.

As for your formula there? A is an acceleration, so you would have to know how quickly the car stopped from 50mph to know its acceleration. Also, you would need to use consistent units - by multiplying the car's mass by its speed, you actually calculated momentum, not force.

Probably the best estimate that we could give using just the information given would be to treat each collision as inelastic, then use momentum conservation to calculate the speed after each separate collision. After collisions with a stationary car, it could be assumed that the car's tires slid all the way to the next collision, and using an assumed frictional coefficient between the rubber and the road, the amount of energy/momentum lost between each collision could be estimated. This would probably not be terribly far off in estimating the way that the collision occurred.

However, for a car to be hit twice, one of two things must be true. Either the driver of car 2 hit car 3 first, then car 1 hit all of them (which is fairly unlikely, as it requires two separate drivers to both cause an accident at the same time), or the collision was not perfectly inelastic. The problem is, predicting the coefficient of restitution (basically, how "bouncy" a collision is) of two cars hitting at 50mph is not something that is easy to do. It is entirely possible that after car 1 hit car 2, car 2 was thrown forwards into car 3, after which car 1 hit both car 2 and car 3 again (resulting in the double impact). However, there's really no way for us to determine if this was actually the case, as there are simply too many variables. A third possibility is that the person claiming to have been hit twice is misremembering - people's memory of events like this is notoriously unreliable.

To sum it up, sadly, this is not a simple problem. Simple physics problems make a lot of assumptions about the way things behave, as without those assumptions, a solution is difficult if not impossible to attain (and thus everything in my post above is simply speculation). A professional accident investigator is much more likely to be able to reconstruct the incident, based on much more information than we have here (for example, the skid marks in the road and the condition of the cars, among other things).

mfb said:
I think it is clear that the driver of car 1 caused an accident by crashing into car 2. It is highly unlikely that two drivers cause independent accidents at the same time
Probably right, but there is another possibility. If driver 2 was too late in applying the brakes and therefore struck car 3 (stationary), car 2 will have stopped very quickly, perhaps taking driver 1 by surprise. This does mean that, in principle, driver 1 was driving too close for the speed, but only by the amount that most drivers (in my view) do. So it's not that unlikely.
Note also the ergonomics of brakes. Assuming all drivers are using footbrakes, not parking brakes, when a shunt occurs the driver in the leading car is thrown backwards relative to the car, releasing the brake. Meanwhile, the driver in the rearward car is thrown forward, braking harder than before. This would contribute to the cars separating after the shunt, rather than coalescing.
For the purposes of the court case, yes indeed, get an accident expert. But I would think it's the existence of a reasonable alternative explanation that's the best hope. She shouldn't need to prove her innocence.

If it's of any help, someone hit from behind and shoved into someone else is never held criminally responsible. It's only by insurance company rules that she would be liable, and that is only for hitting the car in front of her. It's very complicated, but in the long run the instigator (the guy who hit her) is responsible over all. There are several terms for the insurance company policies, none of which I'm allowed to use here.
It's a plain fact that anyone who hits someone else from behind is at fault, unless the front person backed into him/her or an external impulse is applied (as in the guy in the rearmost car). ***-end Charley is always the overall bad guy, because his/her negligence caused the initial impact.

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