# Why do inelastic collisions occur in theoretical calculations?

When solving collision problems related to the conservation of momentum in my applied maths course, the question of whether the collision was elastic or not is often asked. A lot of them time (such as when a car and a truck collide) the collision is inelastic, even though momentum was conserved.

It is then explained to us that the loss of energy was due to a conversion into the form of heat, sound etc. My question is - how is this the case, when these sort of formulae are mostly theoretical, and do not "take into account" sound, heat, friction etc.? I.e. my line of think is that all theoretical collisions that do not consider heat losses would/should be 100% elastic, until we take heat loss into account.

This is the case with many other physics problems, where everything is ideal until we add in calculations to make it more realistic, so why do inelastic collisions occur even in purely theoretical calculations?

• The biggest part of the lost Kinetic Energy in a car collision is the damage done to the car. They are actually designed to take that energy away. To your point you can make the problem more complex by considering heat and sound, but the amount of energy you are talking about there is extremely small compared to the losses due to car deformation. Commented Oct 20, 2020 at 14:25

Collisions always conserve momentum. That' s because forces act in opposite directions. On the other hand, conserving momentum is not enough. There are infinitely many possibilities to conserve it, so a scattering can have many theoretical outcomes.

Adding the energy constraint makes it a solvable problem. According to how much energy you preserve, you have different unique outcomes. So, you need to distinguish between elastic and inelastic scattering, and losses must be included in your qualitative description.

It is then explained to us that the loss of energy was due to a conversion into the form of heat, sound etc.

In an inelastic collision part of the macroscopic kinetic energy of the moving objects is "lost" due to being converted to internal microscopic kinetic energy associated with the random motions of the atoms and molecules of the object and permanent inelastic deformation of the object(s) and the intermolecular friction associated with the deformation.

Part of the kinetic energy is temporarily converted to elastic potential energy (like an ideal spring) which is recovered and converted back to kinetic energy. In a 100% elastic collision no kinetic energy is lost, it is just temporarily converted to elastic potential energy due to elastic deformation of the object (like an ideal spring) and subsequently totally converted back to kinetic energy. In the macroscopic world, there are no perfectly elastic collisions.

The energy lost due to increased molecular motions and the friction associated with permanent deformation manifests itself primarily as an increase in temperature (an partly as sound, as discussed next).

The collision also causes the object to vibrate which in turn causes movement of the surrounding air molecules which in turn manifests itself as sound.

Heat is not an energy form, but energy transfer due solely to temperature difference, and is discussed below.

My question is - how is this the case, when these sort of formulae are mostly theoretical, and do not "take into account" sound, heat, friction etc.? I.e. my line of think is that all theoretical collisions that do not consider heat losses would/should be 100% elastic, until we take heat loss into account.

The losses can be "taken into account", at least theoretically, in formulae associated with increased internal energy (which lumps the effect of increased molecular motion and friction due to permanent deformation), heat transfer, and sound.

Increase in Internal Energy-

$$\Delta KE=\Delta U=mC\Delta T$$

Where $$\Delta KE$$ is the loss of mechanical kinetic energy due to the inelastic collision, $$\Delta U$$ is the increase in internal energy due to increased molecular motion and friction in the collision due to inelastic (permanent) deformation, $$m$$ is the mass of the colliding object, $$C$$ is its specific heat capacity of the material of the object, and $$\Delta T=$$ the increase in temperature of the object due to the inelastic collision.

Heat-

Heat is energy transfer between things due solely to a temperature difference between them.

Following the collision, there will be heat transfer from the object whose temperature was elevated, to the environment, Per Newton's law of cooling

$$Q=hA(T_{obj}-T_{air})$$

where

$$h$$ = the convection heat transfer coefficient of the air (moving relative to the object)

$$A$$ = the convection surface area of the object

$$T$$ = the temperature of the object immediately after the collision

$$T_{air}$$ = the temperature of the environment (air)

Sound-

In a collision, sound energy is a very small part of the total kinetic energy loss. Moreover, sound is itself essentially kinetic energy, in this case the kinetic energy of the surrounding air that has been set into vibration motion by the vibrations of the object from the collision.

The intensity $$I$$, or power per unit area carried by a sound wave is

$$I=\frac{p}{A}$$

And, theoretically

$$I=\frac{(\Delta p)^2}{2ρv_w}$$

where

$$\Delta p=$$ the pressure variation or pressure amplitude

$$v_w$$= the speed of sound in the medium.

$$ρ$$ is the density of the medium (air)

For more details, see

https://courses.lumenlearning.com/physics/chapter/17-3-sound-intensity-and-sound-level/

If none of these losses occur, then the collision would be purely (100%) elastic, such as a collision with an ideal spring.

Hope this helps.