Forest UCM MnAM ElasticCol

An Elastic collision conserves both Momentum and Energy

$\vec{P}_{\mbox{initial}} = \vec{P}_{\mbox{final}}$

AND

$E_{\mbox{initial}} = E_{\mbox{final}}$

Example: problem 3.5

Consider an elastic collision of two equal balls of mass $m$ where one ball has an initial velocity $\vec{v}_1$ and the remaining ball has zero initial velocity.

Determine the angle between the two balls after the collision.

Conservation of momentum

$m_1 \vec{v}_1 + m_2 \vec{v}_2= m_1 \vec{v}_1^{\;\prime} +m_2 \vec{v}_2^{\;\prime}$

$m_1 \vec{v}_1 = m_1 \vec{v}_1^{\;\prime} +m_2 \vec{v}_2^{\;\prime}$: ball 2 has zero velocity

$\vec{v}_1 = \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}$: balls have equal masses

Conservation of energy

$m_1 v_1^2 + m_2 v_2^2 = m_1 \left(v_1^{\prime}\right)^2 + m_2\left(v_2^{\prime}\right)^2$

$v_1^2 = \left(v_1^{\prime}\right)^2 + \left(v_2^{\prime}\right)^2$

If I look at the dot product of the conservation of momentum equation

$\vec{v}_1 = \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}$: balls have equal masses

$\vec{v}_1 \cdot \vec{v}_1 = \left ( \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}\right ) \cdot \left ( \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}\right )$

$v_1^2 = \left(v_1^{\prime}\right)^2 + \left(v_2^{\prime}\right)^2 + 2 \vec{v}_1^{\;\prime} \cdot \vec{v}_2^{\;\prime}$

In order for both the conservation of momentum and energy properties for eleastic collisions to hold

$\vec{v}_1^{\;\prime} \cdot \vec{v}_2^{\;\prime} = 0$

or the balls make an angle of 90 degrees with repect to eachother.

unless one ball has a final velocity of zero

It can happen that the ball with an initial velocity comes to rest after hitting the second ball. To conserve momentum the second ball that was initially at rest must now move with the same momentum as the first ball.

Forest_UCM_MnAM#Elastic_Collision_of_2_bodies