Forest UCM MnAM ElasticCol

An Elastic collision conserves both Momentum and Energy

[math]\vec{P}_{\mbox{initial}} = \vec{P}_{\mbox{final}}[/math]

AND

[math]E_{\mbox{initial}} = E_{\mbox{final}}[/math]

Example: problem 3.5

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

Determine the angle between the two balls after the collision.

Conservation of momentum

[math]m_1 \vec{v}_1 + m_2 \vec{v}_2= m_1 \vec{v}_1^{\;\prime} +m_2 \vec{v}_2^{\;\prime}[/math]

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

[math]\vec{v}_1 = \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}[/math]: balls have equal masses

Conservation of energy

[math]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[/math]

[math]v_1^2 = \left(v_1^{\prime}\right)^2 + \left(v_2^{\prime}\right)^2[/math]

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

[math]\vec{v}_1 = \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}[/math]: balls have equal masses

[math]\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 )[/math]

Forest_UCM_MnAM#Elastic_Collision_of_2_bodies