Practicing the basics
Amy is taking physics this semester, and she has been given leave to ask me questions about it when needed. Apparently they are studying the principles of energy and momentum conservation (i.e., collisions, for which you have to consider both types of conservation in general to make your calculations), though I doubt the professor has explained it quite the same way I did in answering her query. Here's the simplified version of the notes I made for her.
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Amy's problem asks us to determine which does more damage, a semi hitting a motionless car or a car hitting a motionless semi. Intuitively, it seems like the semi should do more damage, but there's a twist -- the problem defines damage peculiarly, saying one is to equate damage with the amount of kinetic energy lost during the collision. People generally don't have an intuition about which collision represents a greater loss of kinetic energy, and even I didn't think about the best way to explain it until I was nearly done sketching these notes out, so generally you want to run through the numbers if you haven't done something similar before. In particular, it is important to note that this question is not asking whether energy is conserved. When someone asks you if energy is conserved, the best answer is still, "If you mean the total of all possible types of energy in the situation you're looking at, then yes, energy is conserved". No, this problem is an investigation into the level of understanding one has for conservation laws and the circumstances under which each one is to be applied.
And exactly how does this problem probe one's understanding of conservation laws? Well, collisions in general fall on a spectrum of elasticity, but we physicists have defined this spectrum in a very odd way. One end of the spectrum is defined as completely* elastic, meaning that the total kinetic energy of all bodies making up the system does not change, while the other end is defined as completely* inelastic, meaning that the bodies are stuck together after the collision. There's even more room for confusion because, when the collision is somewhere in between these two endpoints, the expressions partially elastic, partially inelastic, or simply inelastic are sometimes used. However, there is an easy rule of thumb to avoid becoming confused: if the terms completely elastic or perfectly elastic were not used, then the total kinetic energy of the system was not conserved, making the collision is inelastic. It really is that simple.
Of course, you might have noticed that one end of this spectrum is defined with respect to total kinetic energy of the system while the other is defined with respect to an entirely different quantity, the configuration of the the system. When I first learned about collisions this seemed to illogical to be useful, but it actually makes a lot of sense when one considers the rule of thumb I just described. Some kinetic energy will be given up to the processes that through the processes that occur during this interaction**, whether those processes represent the charge exchange processes in ionospheric plasmas, Rutherford scattering, or the annihilation of matter and antimatter, such as during the operation of LEP. Heck, we have a report of a description of the mass of a proton as mostly the energy involved in binding quarks together, whereas the universe used to be hot enough that quarks roamed freely. It's easy to see that in general the total kinetic energy of the system will not be conserved during a collision. In order to take advantage of this spectrum analogy for talking about collisions, we need a unique state of the system not based upon the initial and final kinetic energies of the system. What's more unique than a bunch of stuff that was separate before sticking together after a collision?
So now we've gone through the nomenclature, and we've established that we can't use conservation of kinetic energy as a tool to make our analysis and solve the problem. What other principles apply? Well, another property that is conserved it momentum (in this case, linear momentum, though in many situations angular momentum will be conserved as well). To the best level anyone anywhere has ever been able to determine, momentum is always conserved. In fact, momentum conservation is so useful a principle in physics that it led Pauli to posit the existence of the neutrino rather than give up the principle. And fortunately in classical physics, momentum and kinetic energy are closely related:
kinetic energy = 1/2 * mass * velocity^2 = momentum^2 / (2 * mass)
If we are given a situation involving completely inelastic collisions, we know immediately that we want the finesse of momentum conservation, not the bludgeon of energy conservation. Even in particle accelerator experiments, where gigantic and elaborate systems of detectors measure the distribution of quantities like temperature and kinetic energy for the particles produced during operation, one must take advantage of momentum conservation to match jets of particles and thus deduce what sort of particles were produced by a given collision. So, let's walk through the problem.
-> The two situations have equal kinetic energy BEFORE the collision.
E_initial_truck = E_initial_car = E_0
-> Define damage to be amount of kinetic energy lost in collision.
-> There are two situations asked about.
1) truck strikes motionless car
2) car strikes motionless truck
-> Note the details.
a) the truck is a semi, so m_truck >> m_car
b) the collisions are completely inelastic since the vehicles are stuck together after the collision
-> Relate kinetic energy to momentum.
E = (1/2)*m*v^2 = p^2 / (2*m) since p = m * v
1) Consider the truck hitting the motionless car.
-> Apply momentum conservation: p_initial = p_final = p.
p = m_truck * v_truck_initial = (m_truck m_car) * v_1_final, so
v_1_final = m_truck * v_truck_initial / (m_truck m_car).
-> Find the difference between the initial and the final energies.
E_1_final = 1/2 * (m_truck m_car)*v_1_final^2, so
E_1_final = 1/2 * m_truck * v_truck_initial^2 * [m_truck / (m_truck m_car)], giving
E_1_final = E_0 * [m_truck / (m_truck m_car)].
E_1_lost = E_0 - E_1_final = E_0 * (1 - m_truck / (m_truck m_car))
2) Consider the car hitting the motionless truck.
-> Apply momentum conservation: p_initial = p_final = p.
p = m_car * v_car_initial = (m_truck m_car) * v_2_final, so
v_2_final = m_car * v_car_initial / (m_truck m_car)
-> Find the difference between the initial and the final energies.
Following the above procedure, we get the same form but with
E_2_final replacing E_1_final everywhere and m_car replacing m_truck
everywhere. Thus,
E_2_lost = E_0 * (1 - m_car / (m_truck m_car))
3) Make a conclusion.
m_car is smaller than m_truck, so E_2_lost > E_1_lost. Given our wacky definition for damage, more damage is done when the car hits a parked truck than when a truck hits a parked car.
At this point we have a conclusion, but we are not actually done with the problem. It remains to think hard and determine if this prediction matches what one observes. This is where I told Amy about the description the prof gave when I took freshman physics oh so many years ago. His physics fable involved a bowling ball and a BB pellet. The case of the car hitting the semi, that's like a bowling ball sitting there, minding its own business, when the BB comes out of nowhere hand. The bowling ball is much more massive than the BB, so it looks around a bit to see what just happened but mostly just keeps sitting there, doing exactly what it was doing before. The speed of the bowling ball plus BB is only slightly greater than whatever speed the bowling ball had before the impact, and this speed is much smaller than the speed the BB had before since momentum conservation tells us how the speeds are all related. Essentially ALL of the kinetic energy of the system is lost since the BB is embedded in the bowling ball. The only reason that the bowling ball moves at all is that momentum must be conserved, and its moving very, very slowly compared to how fast the BB was moving.
On the other hand, when the semi hits the parked car, that's like the case where the bowling ball is rolling along and runs over the BB, picking it up as it goes. The bowling ball is so much more massive than the BB that even though it has given up some momentum to the BB, the total kinetic energy of the system is almost the same as it was before the bowling ball trampled the BB. The change in the total kinetic energy is a lot smaller for this case than it is for the other case, so going back to that odd definition of the damage done during a collision, the semi does less damage when hitting a parked car than the a car does when hitting a parked semi.
This is a tricksy conclusion because intuitively a person not trained to think like a physicist could easily expect the semi to deal more damage. In fact, I immediately noticed that the definition for damage didn't seem to be a useful way to assess damage and therefore guessed that the car would deal out more carnage, but I was led astray by intuition while writing the notes up. It was only when I was double checking the notes before cutting and pasting the notes into the gaim window and added the line that says "m_car is smaller than m_truck" as a reminder that made me change my mind back to the correct conclusion. And indeed this is exactly the point of giving such a peculiar definition for damage, because the author of the problem really want one to think about and internalize these principles and this way of thinking.
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* The word "completely" is often replaced with the word "perfectly" or other similar-sounding words, but the definitions for the endpoints of this nomenclatural spectrum are always the same.
** In the version of the air table-and-hockey puck lab that the Rice honors physics class uses to illustrate these ideas, one source of losses is that the velcro rings used to make the hockey pucks stick together, especially if their velocities before the collision do not lie on the line connecting the center of the two pucks. In this case a torque arises that transfers energy to rotational motion. In addition, the velcro rings can slip, transferring energy to oscillations or friction, or the rings can just come straight off, allowing god knows what else to happen, such as hitting the surface of the table and therefore changing velocities.
----------
Amy's problem asks us to determine which does more damage, a semi hitting a motionless car or a car hitting a motionless semi. Intuitively, it seems like the semi should do more damage, but there's a twist -- the problem defines damage peculiarly, saying one is to equate damage with the amount of kinetic energy lost during the collision. People generally don't have an intuition about which collision represents a greater loss of kinetic energy, and even I didn't think about the best way to explain it until I was nearly done sketching these notes out, so generally you want to run through the numbers if you haven't done something similar before. In particular, it is important to note that this question is not asking whether energy is conserved. When someone asks you if energy is conserved, the best answer is still, "If you mean the total of all possible types of energy in the situation you're looking at, then yes, energy is conserved". No, this problem is an investigation into the level of understanding one has for conservation laws and the circumstances under which each one is to be applied.
And exactly how does this problem probe one's understanding of conservation laws? Well, collisions in general fall on a spectrum of elasticity, but we physicists have defined this spectrum in a very odd way. One end of the spectrum is defined as completely* elastic, meaning that the total kinetic energy of all bodies making up the system does not change, while the other end is defined as completely* inelastic, meaning that the bodies are stuck together after the collision. There's even more room for confusion because, when the collision is somewhere in between these two endpoints, the expressions partially elastic, partially inelastic, or simply inelastic are sometimes used. However, there is an easy rule of thumb to avoid becoming confused: if the terms completely elastic or perfectly elastic were not used, then the total kinetic energy of the system was not conserved, making the collision is inelastic. It really is that simple.
Of course, you might have noticed that one end of this spectrum is defined with respect to total kinetic energy of the system while the other is defined with respect to an entirely different quantity, the configuration of the the system. When I first learned about collisions this seemed to illogical to be useful, but it actually makes a lot of sense when one considers the rule of thumb I just described. Some kinetic energy will be given up to the processes that through the processes that occur during this interaction**, whether those processes represent the charge exchange processes in ionospheric plasmas, Rutherford scattering, or the annihilation of matter and antimatter, such as during the operation of LEP. Heck, we have a report of a description of the mass of a proton as mostly the energy involved in binding quarks together, whereas the universe used to be hot enough that quarks roamed freely. It's easy to see that in general the total kinetic energy of the system will not be conserved during a collision. In order to take advantage of this spectrum analogy for talking about collisions, we need a unique state of the system not based upon the initial and final kinetic energies of the system. What's more unique than a bunch of stuff that was separate before sticking together after a collision?
So now we've gone through the nomenclature, and we've established that we can't use conservation of kinetic energy as a tool to make our analysis and solve the problem. What other principles apply? Well, another property that is conserved it momentum (in this case, linear momentum, though in many situations angular momentum will be conserved as well). To the best level anyone anywhere has ever been able to determine, momentum is always conserved. In fact, momentum conservation is so useful a principle in physics that it led Pauli to posit the existence of the neutrino rather than give up the principle. And fortunately in classical physics, momentum and kinetic energy are closely related:
kinetic energy = 1/2 * mass * velocity^2 = momentum^2 / (2 * mass)
If we are given a situation involving completely inelastic collisions, we know immediately that we want the finesse of momentum conservation, not the bludgeon of energy conservation. Even in particle accelerator experiments, where gigantic and elaborate systems of detectors measure the distribution of quantities like temperature and kinetic energy for the particles produced during operation, one must take advantage of momentum conservation to match jets of particles and thus deduce what sort of particles were produced by a given collision. So, let's walk through the problem.
-> The two situations have equal kinetic energy BEFORE the collision.
E_initial_truck = E_initial_car = E_0
-> Define damage to be amount of kinetic energy lost in collision.
-> There are two situations asked about.
1) truck strikes motionless car
2) car strikes motionless truck
-> Note the details.
a) the truck is a semi, so m_truck >> m_car
b) the collisions are completely inelastic since the vehicles are stuck together after the collision
-> Relate kinetic energy to momentum.
E = (1/2)*m*v^2 = p^2 / (2*m) since p = m * v
1) Consider the truck hitting the motionless car.
-> Apply momentum conservation: p_initial = p_final = p.
p = m_truck * v_truck_initial = (m_truck m_car) * v_1_final, so
v_1_final = m_truck * v_truck_initial / (m_truck m_car).
-> Find the difference between the initial and the final energies.
E_1_final = 1/2 * (m_truck m_car)*v_1_final^2, so
E_1_final = 1/2 * m_truck * v_truck_initial^2 * [m_truck / (m_truck m_car)], giving
E_1_final = E_0 * [m_truck / (m_truck m_car)].
E_1_lost = E_0 - E_1_final = E_0 * (1 - m_truck / (m_truck m_car))
2) Consider the car hitting the motionless truck.
-> Apply momentum conservation: p_initial = p_final = p.
p = m_car * v_car_initial = (m_truck m_car) * v_2_final, so
v_2_final = m_car * v_car_initial / (m_truck m_car)
-> Find the difference between the initial and the final energies.
Following the above procedure, we get the same form but with
E_2_final replacing E_1_final everywhere and m_car replacing m_truck
everywhere. Thus,
E_2_lost = E_0 * (1 - m_car / (m_truck m_car))
3) Make a conclusion.
m_car is smaller than m_truck, so E_2_lost > E_1_lost. Given our wacky definition for damage, more damage is done when the car hits a parked truck than when a truck hits a parked car.
At this point we have a conclusion, but we are not actually done with the problem. It remains to think hard and determine if this prediction matches what one observes. This is where I told Amy about the description the prof gave when I took freshman physics oh so many years ago. His physics fable involved a bowling ball and a BB pellet. The case of the car hitting the semi, that's like a bowling ball sitting there, minding its own business, when the BB comes out of nowhere hand. The bowling ball is much more massive than the BB, so it looks around a bit to see what just happened but mostly just keeps sitting there, doing exactly what it was doing before. The speed of the bowling ball plus BB is only slightly greater than whatever speed the bowling ball had before the impact, and this speed is much smaller than the speed the BB had before since momentum conservation tells us how the speeds are all related. Essentially ALL of the kinetic energy of the system is lost since the BB is embedded in the bowling ball. The only reason that the bowling ball moves at all is that momentum must be conserved, and its moving very, very slowly compared to how fast the BB was moving.
On the other hand, when the semi hits the parked car, that's like the case where the bowling ball is rolling along and runs over the BB, picking it up as it goes. The bowling ball is so much more massive than the BB that even though it has given up some momentum to the BB, the total kinetic energy of the system is almost the same as it was before the bowling ball trampled the BB. The change in the total kinetic energy is a lot smaller for this case than it is for the other case, so going back to that odd definition of the damage done during a collision, the semi does less damage when hitting a parked car than the a car does when hitting a parked semi.
This is a tricksy conclusion because intuitively a person not trained to think like a physicist could easily expect the semi to deal more damage. In fact, I immediately noticed that the definition for damage didn't seem to be a useful way to assess damage and therefore guessed that the car would deal out more carnage, but I was led astray by intuition while writing the notes up. It was only when I was double checking the notes before cutting and pasting the notes into the gaim window and added the line that says "m_car is smaller than m_truck" as a reminder that made me change my mind back to the correct conclusion. And indeed this is exactly the point of giving such a peculiar definition for damage, because the author of the problem really want one to think about and internalize these principles and this way of thinking.
----------
* The word "completely" is often replaced with the word "perfectly" or other similar-sounding words, but the definitions for the endpoints of this nomenclatural spectrum are always the same.
** In the version of the air table-and-hockey puck lab that the Rice honors physics class uses to illustrate these ideas, one source of losses is that the velcro rings used to make the hockey pucks stick together, especially if their velocities before the collision do not lie on the line connecting the center of the two pucks. In this case a torque arises that transfers energy to rotational motion. In addition, the velcro rings can slip, transferring energy to oscillations or friction, or the rings can just come straight off, allowing god knows what else to happen, such as hitting the surface of the table and therefore changing velocities.