# Conservation of energy in elastic collisions

We can say that collisions happen when two or more bodies try to occupy the same position at the same time, as when two cars collide at an intersection. It is possible to find collisions of different dimensions, that is, we can have microscopic collisions, such as collisions that occur between atoms, and macroscopic collisions, such as collisions between two cars, etc.

In elastic collisions, the total mechanical energy of the system always has the same value. If one of the parts increases its mechanical energy, some other part will have its energy reduced.

*E _{total mechanics before collision} =E _{total mechanics after collision}*

It is important to emphasize that the total energy, that is, the sum of all the energies involved in the system, is conserved. Therefore, in most situations involving elastic collisions, the potential energy of the system remains the same, and therefore, the only form of energy that we must take into account is kinetic energy.

Therefore, we have:

*E _{Ctotal before} = E _{Ctotal after}*

For the above equation we are assuming only two objects involved in the collision.

Conservation of mechanical energy only applies to elastic collisions, but conservation of momentum applies to any type of collision. Assuming a collision in one dimension, we have:

*P _{total before} = P _{total after}*

*m _{1} .v _{1 before} +m _{2} .v _{2 before} =m _{1} .v _{1 after} )+m _{2} .v _{2 after}*

If the masses of the objects and the velocities before the collision are known, we have two equations with two unknowns. To simplify the solution of the two conservation equations we can do the following:

*m _{1} (v _{1 before} -v _{1 after} )=m _{2} (v _{2 after} -v _{2 before} )*

*m _{1} (v _{1 }*

^{2 }

_{before}-v_{1 }

^{2 }

_{after})=m_{2}*(v*

_{2 }^{2 }

_{after}-v_{2 }^{2 }

_{before})Dividing the second equation by the first, we get:

*v _{1 before} +v _{1 after} =v _{2 after} -v _{2 before}*

Which can also be written as follows:

*v _{1 before} -v _{2 before} =v _{2 after} -v _{1 after}*

This equation expresses that the relative speed of approach is equal to the relative speed of departure.