Discuss elastic collision in two dimensions.

(N/A) As shown in the figure,suppose a ball of mass $m_{1}$ moving in the $X$-direction with speed $v_{1i}$ collides elastically with a stationary ball of mass $m_{2}$.
After the collision,these balls move in directions making angles $\theta_{1}$ and $\theta_{2}$ with the $X$-axis with velocities $v_{1f}$ and $v_{2f}$ respectively.
Momentum is conserved in the collision,so the total momentum before collision equals the total momentum after collision.
Taking $X$-components of momentum:
$m_{1} v_{1i} = m_{1} v_{1f} \cos \theta_{1} + m_{2} v_{2f} \cos \theta_{2} \quad \dots (1)$
Taking $Y$-components of momentum:
$0 = m_{1} v_{1f} \sin \theta_{1} - m_{2} v_{2f} \sin \theta_{2} \quad \dots (2)$
Since the collision is elastic,kinetic energy is conserved:
$\frac{1}{2} m_{1} v_{1i}^{2} = \frac{1}{2} m_{1} v_{1f}^{2} + \frac{1}{2} m_{2} v_{2f}^{2} \quad \dots (3)$
Here,we have three independent equations $(1)$,$(2)$,and $(3)$. Typically,$m_{1}, m_{2},$ and $v_{1i}$ are known,while the four variables $v_{1f}, v_{2f}, \theta_{1},$ and $\theta_{2}$ are unknown. To solve for all unknowns,at least one of these four quantities must be known,as three equations can only determine three unknown quantities.