# Elastic and inelastic collisions

Elastic and inelastic collisions are interactions between bodies in which one exerts force on the other, being classified according to the conservation of energy.

Collisions are interactions between bodies in which one exerts force on the other. See what are the inherent characteristics of **elastic and inelastic collisions.**

**elastic collisions**

The collision is called elastic when there is conservation of energy and linear momentum of the bodies involved. The main characteristic of this type of collision is that, after the collision, the velocity of the particles changes direction, but the relative velocity between the two bodies remains the same. To understand better, look at the example in the figure:

Velocity of bodies A and B before and after an elastic collision

We can see in the figure above that, after the collision, the spheres began to move in the opposite direction to what they had before colliding.

Let us now obtain the equations for the kinetic energy and for the linear momentum:

As previously mentioned, in this type of collision, energy and momentum are conserved. This conservation can be described by the equations:

- For conservation of linear momentum:

Q _{i} = Q _{f} —> m _{A} . V _{IA} + m _{B} . V _{IB} = m _{A} . V _{FA} + m _{B} . V _{FB}

- For the conservation of kinetic energy:

EI = EF —> __1__ m _{A} . V _{IA2} + __1__ mB . _{_ }V _{IB2} = __1__ mA . _{_ }V _{FA2} + __1__ mB . _{_ }V _{FB2
} 2 2 2 2

Being that:

m _{A} in _{B} are the masses of bodies A and B respectively;

V _{I} is the initial velocity;

V _{F} is the final speed.

**inelastic collisions**

If, when a collision occurs, there is no conservation of kinetic energy, it is called an inelastic collision. In this type of collision, the energy can be transformed into another form, for example, into thermal energy, causing the temperature of the objects that collided to increase. In this way, only linear momentum is conserved.

Inelastic collisions can be classified in two ways: perfectly inelastic and partially inelastic.

**Perfectly inelastic collisions:** when the maximum loss of kinetic energy occurs. After this type of collision, the objects remain united as if they were a single body with a mass equal to the sum of the masses before the collision. See the figure:

After a perfectly inelastic collision, the two objects move together in the same direction as if they were a single object.

As previously mentioned, in this case, only the conservation of linear momentum occurs. We can get an expression for the final velocity V _{F} of objects. See the equations below:

Q _{i} = Q _{f} —> m _{A} . V _{IA} + m _{B} . V _{IB} = (m _{A} + m _{B} ) V _{F}

Isolating V _{F} , we have:

V _{F} = __m ___{A}__ . V ___{IA}__ + m ___{B}__ . V ___{IB
} m _{A} + m _{B}

**Partially inelastic collisions:** only part of the kinetic energy is conserved so that the final energy is less than the initial energy. They make up the majority of collisions that occur in nature. In this case, after the collision, the particles separate, and the final relative velocity is lower than the initial one. Look at the figure:

After a partially inelastic collision, the spheres move away with a relative speed different from the approach speed.

The figure above shows the behavior of two spheres before and after a partially inelastic collision. To better understand, we use numerical values for the velocities. The relative speed before collision is given by the difference between the two speeds:

V _{rel} = V _{IA} – V _{IB}

Substituting the values, we have:

Vrel _{=} 6 – (-4) = 10 m/s

After the collision, we have the following situation:

V _{rel} = V _{FA} – V _{FB}

V _{rel} = 3 – (- 4) = 7m/s

We can see that the relative velocity before the collision is different from the relative velocity after the collision. This is what characterizes this collision as partially inelastic, but it can also be called partially elastic.