0804 Elastic Collision

Collision in whichbodies collide and seperate, the velocties are exchanged if purely elastic, where both $\vec P$ and $KE$ are conserved. Examples: billizard balls.
For any two masses, $m_1, m_2$. Let the momentum before and after collision be $\vec P_i$ and $\vec P_f$, the kinetic energy before and after collision be $KE_i$ and $KE_f$

$\begin{aligned} \vec P_i &= \vec P_f\\ m_1v_{1i} + m_2v_{2i} &= m_1v_{1f} + m_2v_{2f}\\ KE_i &= KE_f\\ \frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 &= \frac{1}{2}m_1 v_{2i}^2 + \frac{1}{2}m_2 v_{2f}^2\\ \Downarrow\\ v_{1f} &= \frac{m_1-m_2}{m_1+m_2} \cdot v_{1i}\ + \frac{2m_2}{m_1+m_2}\cdot v_{2i} \\ v_{2f} &= \frac{2m_1}{m_1+m_2}\cdot v_{1i} + \frac{m_2-m_1}{m_1+m_2}\cdot v_{2i} \end{aligned}$

Exercises

46, 48
(8.46) A 0.152 kg glider is moving to the right on a frictionless, horizontal air track with a speed of 0.830 m/s. It has a head-on collision with a 0.307 kg glider that is moving to the left with a speed of 2.30 m/s. Suppose the collision is elastic.
(a) Find the magnitude and direction of the final velocity of the 0.152 kg glider.
(b) Find the magnitude and direction of the final velocity of the 0.307 kg glider.

(8.48) A 10.0 g marble slides to the left with a velocity of magnitude 0.400 m/s on the frictionless, horizontal surface of an icy New York sidewalk and has a head-on, elastic collision with a larger 30.0 g marble sliding to the right with a velocity of magnitude 0.200 m/s. Let $+x$ be to the right. (Since the collision is head-on, all the motion is along a line.)