# Stopping Distance in Conservation of Momentum Investigation

In a set of old textbook materials from Merrill Physics from the mid 90s, there is a page on Fitch Inertial Barriers put near construction zones. Three cars are going to collide into a set of these barriers.

Car 1 has a mass of 910kg and a velocity of 82km/h
Car 3 has a mass of 1200kg and a velocity of 50km/h.
Car 5 has a mass of 1500kg and a velocity of 40km/h.

The other two cars are inconsequential to this discussion.

There are two questions in the exercise which have tripped me up.

Questions

1. Which automobile will penetrate most deeply into the array of barriers?

2. Between cars 3 and 5, which occupants would be most likely to sustain serious injuries?

My thoughts
The one which penetrates the deepest is not obvious to me, although the answer key says the one with the highest initial momentum will penetrate the deepest. How can we tell this from the setup? Must it be the case that the forces on each car are identical due to hitting the same barriers? Then that would make the car with the largest momentum take the longest time and hopefully go the farthest distance.

While thinking about this, I made a short table of values of varying distances travelled when using accelerations and time that multiply to the same number (hence, having the same change in velocity). it's very clear that distance travelled does not stay constant as a and t are varied. Why can the same not be true here, that two cars with the same initial momenta (and even the same mass, if we want) can have different values for a and t which wildly change how far they travel as they hit the barriers?

This seems to tie in to the suggested answer for #2, which is that both cars sustain the same damage to the people due to starting with the same momenta. How do we properly measure this? Is impulse the best measure of injuries sustained? It seems that a high acceleration is damaging, but only if sustained long enough.

I suppose what I'm hoping for is some sort of derivation that will clarify these ideas or some sort of assumption or definition I am missing or do not grasp fully.

• Have you figured the kinetic energy of each car? Nov 28, 2020 at 4:27
• en.wikipedia.org/wiki/Kinetic_energy#Newtonian_kinetic_energy Nov 28, 2020 at 4:30
• @AdrianHoward The kinetic energies of the cars decrease from 1 to 3 to 5, so if I assume the force is constant for all the vehicles, then car 1 drives the farthest and car 5 drives the shortest, with car 3 in-between, using Fs = 1/2 m v^2. That still leaves me wondering if that is a fair assumption I can (or even must) use to answer these questions. Nov 28, 2020 at 18:09
• What does the textbook say about Fitch Inertial Barriers? Doesn't this explain how to tackle the question? Nov 29, 2020 at 4:33
• @sammygerbil Unfortunately, the book says nothing about it. It's just a side worksheet that is an attempt to tie momentum to a real-world scenario. Dec 4, 2020 at 0:35

You are probably looking at this worksheet. It provides no explanation as to how the impact barrier works and therefore on what assumptions and principle(s) any calculations should be made.

There are two types of impact attenuator : compression attenuators which stay in place, deform by crumpling like a concertina, and absorb kinetic energy in an inelastic collision; and inertial attenuators which don't deform significantly but move along with the vehicle after an elastic collision. Fitch Barrier Arrays are the latter type. After coming into contact with the vehicle the barrels move along with it, so that effectively the mass of the vehicle is gradually increased, slowing it down. As suggested, the conservation of momentum principle is the one to use. (See pp 7-8 of this document.)

This model reduces the speed of the vehicle but cannot stop it altogether. The linked document states that the Fitch barrels are used to reduce speed to about 15kph then collision with a final fixed hard barrier halts the vehicle. Such a hard barrier is shown in the worksheet. The barrels between the vehicle and hard barrier get deformed, extending the impact time and reducing forces on the occupant(s).

Collision with the barrels conserves momentum (according to the model used here) but KE is reduced by the factor $$m/(m+M)$$ where $$m, M$$ are the masses of the car and barrels respectively. The remaining KE is dissipated in the collision with the hard barrier.

The depth to which a vehicle penetrates into the array increases with its momentum before impact, because a larger mass of barrels is needed to reduce a larger momentum. So question 1 is easily settled by a comparison of momentum.

If we had been dealing with a compression attenuator instead of an inertial type, the penetration depth would increase with kinetic energy before impact.

The severity of injury to occupants depends on the highest forces on their bodies, which depends on the maximum deceleration. This comparison is less obvious.

The reason cars 3 & 5 are being compared becomes apparent : they have the same momentum before impact, so they will penetrate to the same depth. This also means that the manner in which the deceleration varies with time during impact will be the same : one impact is a slowed-down version of the other. To answer question 2 we can compare average deceleration. Car 3 has the higher initial speed, so its average deceleration over the same distance will be higher.

If the momenta of cars 3 & 5 had not been the same a more complex calculation would have been required.

My answer to question 2 contradicts the suggested answer. I suspect that the suggested answers were not provided by the same person who devised the worksheet. The explanation given is not convincing. This is a sign that the person who wrote it did not understand what he/she was doing.

• Thank you for such a thorough breakdown.tge only thing I do not grasp yet from you is how the increase in mass of the car by sticking to barriers can reduce it's speed to zero without losing at least some amount of energy. Dec 5, 2020 at 13:05
• @toothandsticks This model reduces speed but not KE, and cannot reduce the speed to zero. The linked document states that the fitch barrels are used to reduce speed to about 15kph then collision with a final fixed hard barrier halts the vehicle. The barrels between the vehicle and hard barrier get deformed, extending impact time and reducing forces on occupant, and dissipating the KE. Dec 6, 2020 at 1:30
• @toothandsticks Please upvote my answer or select as "best answer" if you are satisfied with it. If not, please let me know if there's anything else which I can explain. Dec 8, 2020 at 5:25
• Thank you for your thorough and swift responses. Dec 10, 2020 at 13:53