Inelastic Collisions

Inelastic Collisions

So the total momentum before an inelastic collisions is the same as after the collision. But the total kinetic energy before and after the inelastic collision is different. Of course this does not mean that total energy has not been conserved, rather the energy has been transformed into another type of energy.

As a rule of thumb, inelastic collisions happen when the colliding objects are distorted in some way. Usually they change their shape. The modification of the shape of an object requires energy and this is where the “missing” kinetic energy goes. A classic example of an inelastic collision is a motor car accident. The cars change shape and there is a noticeable change in the kinetic energy of the cars before and after the collision. This energy was used to bend the metal and deform the cars. Another example of an inelastic collision is shown in the figure below.

Asteroid moving towards the Moon.

An asteroid is moving through space towards the Moon. Before the asteroid crashes into the Moon, the total momentum of the system is:

\[\vec{p}_{Ti}=\vec{p}_{iM}+\vec{p}_{ia}\]

The total kinetic energy of the system is:

\[K{E}_{i}=K{E}_{iM}+K{E}_{ia}\]

When the asteroid collides inelastically with the Moon, its kinetic energy is transformed mostly into heat energy. If this heat energy is large enough, it can cause the asteroid and the area of the Moon's surface that it hits, to melt into liquid rock. From the force of impact of the asteroid, the molten rock flows outwards to form a crater on the Moon.

After the collision, the total momentum of the system will be the same as before. But since this collision is inelastic, (and you can see that a change in the shape of objects has taken place), total kinetic energy is not the same as before the collision.

Momentum is conserved:

\[\vec{p}_{Ti}=\vec{p}_{Tf}\]

But the total kinetic energy of the system is not conserved:

\[K{E}_{i}\ne K{E}_{f}\]

Example: An Inelastic Collision

Question

Consider the collision of two cars. Car 1 is at rest and Car 2 is moving at a speed of \(\text{2}\) \(\text{m·s$^{-1}$}\) to the left. Both cars each have a mass of \(\text{500}\) \(\text{kg}\). The cars collide inelastically and stick together. What is the resulting velocity of the resulting mass of metal?

Step 1: Draw a rough sketch of the situation

Step 1: Determine how to approach the problem

We are given:

• mass of car 1, \({m}_{1}=500\text{kg}\)

• mass of car 2, \({m}_{2}=500\text{kg}\)

• initial velocity of car 1, \(\vec{v}_{i1}=0\text{m}{\text{s}}^{-1}\)

• initial velocity of car 2, \(\vec{v}_{i2}=2\text{m}{\text{s}}^{-1}\) to the left

• the collision is inelastic

All quantities are in SI units. We are required to determine the final velocity of the resulting mass, \(\vec{v}_{f}\).

Since the collision is inelastic, we know that

• momentum is conserved, \({m}_{1}\vec{v}_{i1}+{m}_{2}\vec{v}_{i2}={m}_{1}\vec{v}_{f1}+{m}_{2}\vec{v}_{f2}=\left({m}_{1}+{m}_{2}\right)\vec{v}_{f}\)

• kinetic energy is not conserved

Step 2: Choose a frame of reference

Choose to the left as positive.

Step 3: Solve problem

So we must use conservation of momentum to solve this problem.

\begin{align*} \vec{p}_{Ti}& = \vec{p}_{Tf}\\ \vec{p}_{i1}+\vec{p}_{i2}& = \vec{p}_{f}\\ {m}_{1}\vec{v}_{i1}+{m}_{2}\vec{v}_{i2}& = \left({m}_{1}+{m}_{2}\right)\vec{v}_{f}\\ \left(500\right)\left(0\right)+\left(500\right)\left(2\right)& = \left(500+500\right)\vec{v}_{f}\\ 1000& = 1000\vec{v}_{f}\\ \vec{v}_{f}& = 1\text{m·s$^{-1}$} \end{align*}

Therefore, the final velocity of the resulting mass of cars is \(\text{1}\) \(\text{m·s$^{-1}$}\) to the left.