Elastic Collisions in One Dimension

An elastic collision is a collision between two or more bodies in which the total kinetic energy of the bodies before the collision is equal to the total kinetic energy of the bodies after the collision. An elastic collision will not occur if kinetic energy is converted into other forms of energy. It important to understand how elastic collisions work, because atoms often undergo essentially elastic collisions when they collide. On the other hand, molecules do not undergo elastic collisions when they collide . In this atom we will review case of collision between two bodies.

The mathematics of an elastic collision is best demonstrated through an example. Consider a first particle with mass $m_{1}$ and velocity $v_{1i}$ and a second particle with mass $m_{2}$ and velocity $v_{2i}$. If these two particles collide, there must be conservation of momentum before and after the collision. If we know that this is an elastic collision, there must be conservation of kinetic energy by definition. Therefore, the velocities of particles 1 and 2 after the collision ($v_{1f}$ and $v_{2f}$ respectively) will be related to the initial velocities by:

$\frac{1}{2}m_1\cdot v_{1i}^2+\frac{1}{2}m_2\cdot v_{2i}^2=\frac{1}{2}m_1\cdot v_{1f}^2+\frac{1}{2}m_2\cdot v_{2f}^2$ (due to conservation of kinetic energy)

and

$m_1\cdot v_{1i}+m_2\cdot v_{2i}=m_1\cdot v_{1f}+m_2\cdot v_{2f}$ (due to conservation of momentum).

Since we have two equations, we are able to solve for any two unknown variables. In our case, we will solve for the final velocities of the two particles.

By grouping like terms and canceling out the ½ terms, we can rewrite our conservation of kinetic energy equation as:

$m_1\cdot (v_{1i}^2-v_{1f}^2) = m_2\cdot (v_{2f}^2-v_{2i}^2)$. (Eq.1)

By grouping like terms from our conservation of momentum equation we can find:

$m_1\cdot (v_{1i}-v_{1f}) = m_2\cdot (v_{2f}-v_{2i})$. (Eq. 2)

If we then divide Eq. 1 by Eq. 2 and perform some cancelations we will find:

$v_{1i} + v_{1f} = v_{2f} + v_{2i}$. (Eq. 3)

We can solve for $v_{1f}$ as:

$v_{1f} = v_{2f} + v_{2i}-v_{1i}$. (Eq. 4)

At this point we see that $v_{2f}$ is still an unknown variable. So we can fix this by plugging Eq. 4 into our initial conservation of momentum equation. Our conservation of momentum equation with Eq. 4 substituted in looks like:

$m_1\cdot v_{1i}+m_2\cdot v_{2i}=m_1\cdot(v_{2f} + v_{2i}-v_{1i})+m_2\cdot v_{2f}$. (Eq.5)

After doing a little bit of algebra on Eq. 5 we find:

$v_{2f} =\frac{2\cdot m_1}{(m_2+m_1)}v_{1i} +\frac{(m_2-m_1)}{(m_2+m_1)}v_{2i}$. (Eq.6)

At this point we have successfully solved for the final velocity of the second particle. We still need to solve for the velocity of the first particle, so let us do that by plugging Eq. 6 into Eq. 4.

$v_{1f} = [\frac{2\cdot m_1}{(m_2+m_1)}v_{1i} +\frac{(m_2-m_1)}{(m_2+m_1)}v_{2i}] + v_{2i}-v_{1i}$. (Eq. 7)

After performing some algebraic manipulation of Eq. 7, we finally find:

$v_{1f} =\frac{(m_1-m_2)}{(m_2+m_1)}v_{1i}+\frac{2\cdot m_2}{(m_2+m_1)}v_{2i}$. (Eq. 8)

Elastic Collision of Two Unequal Masses

In this animation, two unequal masses collide and recoil.