# Heat generated by collision

Suppose there are two objects: A = m1 and B = m2 . A is travelling at a constant velocity v toward B and collides inelastically: is there a way in which I could determine the energy which is dissipated as heat? What other variables I should know before it can be calculated?

Edit: I only know the values before the collision, for example the initial velocities $V_A, V_B$, initial temperatures, surface areas, etc.

Imagine that I am running at speed V through a cloud of flies (just an example), can determine how much energy is dissipated through heat without actually having to run through the cloud of flies and measuring the change in variables?

What other variables I should know before it can be calculated? : I only know the values before the collision, for example the initial velocities VA,VB, initial temperatures, surface areas, etc.

You need to know the $C_R$ coefficient of restitution, CR is the coefficient of restitution if it is 1 we have an elastic collision, if it is 0 we have a perfectly inelastic collision, see below.

Else, you must know the result of the collision and subtract to the initial energy of the system its final energy. But not all missing energy must have been transformed to thermal energy, some energy may be dissipated through sound etc.

• @JoshuaLin , Surely it depends on materials, but also on other factors, like speed etc. Oct 1, 2014 at 7:22
• @JoshuaLin, if the impact is at 2 Km/h, do you think heat can be the same as it takes place at 300 Km/h? Oct 1, 2014 at 7:26
• I am not an expert on materials, but I suppose every material has a different curve of response and the increase must not necessarily be linear, also the percent can vary. Oct 1, 2014 at 7:28
• @JoshuaLin See the elastic limit or yield point for materials. For stress below this point, the material may behave mostly elastically, which reduces the heat generated. Stress past that point (perhaps due to a high-speed collision) will cause permanent deformation and increased energy loss that becomes heat. The COR is not fixed for a material. It depends on the specific interaction. It may be different for different impact speeds. Oct 1, 2014 at 7:56
• I guess it'd be pretty hard to calculate the heat energy dispersed by a spacecraft driving at high speeds through cosmic dust then huh... Thanks for all the help though Oct 1, 2014 at 7:58

Fundamentally, it comes down to the conservation of energy, since we already know the initial velocity of both objects, by simply measuring the velocities of the objects just after the collision (say ${v_1}$ for object 1 and $v_2$ for object 2), we can calculate the amount of energy converted to heat by simply using, $${ m_1v^2 = m_1v_1^2 + m_2v_2^2 + Heat}$$

For this I have assumed that during the collision, no extra energy is introduced into the system or lost from the system. For example, in real life, if two cars collide, and the engine of car 1 explodes, this introduces energy into the system which must be handled separately.

Thus for this, we can modify the conservation of energy as, $${ m_1v^2 + Energy~introduced~into~the~system = m_1v_1^2 + m_2v_2^2 + Heat+Energy~lost~by~the~system}$$

EDIT: If only the initial conditions are know, although it is theoretically possible to find the velocity of the object before collision, practically, this is near impossible for complex systems such as the example you have pointed out. However for simple systems, it fundamentally works down that all the forces during a collision are electrostatic in nature (rarely gravitational), if it is possible to list all the forces between the individual molecules/objects, then yes, it is possible.

EDIT 2: As for the equation, it would be a second order ODE if you factor in all the force components (as F = m * a = K/(r^2) ). As it is a differential equation factoring in n-forces it does not have a simple equation form which can be written out.

• Yes that's true, but the problem I face is that I don't know the speed after the collision, should have clarified. Like for friction, we know that friction depends on the coefficient of static/kinetic friction, which varies from material to material. I'm sort of searching for an equation that given initial variables, such as velocity, mass, surface area, material, coefficient of friction, will output energy dispersed in heat. Oct 1, 2014 at 6:55
• At a minimum then you would also need the coefficient of restitution, which is the ratio of the speeds after and before the collision. A COR of 1 would have no energy dissipated as heat. A COR of 0 would turn all the KE into heat (when measured in the frame with the center of mass at rest) Oct 1, 2014 at 7:01