# Collision in Two Dimensions

**Elastic Collision in Two Dimensions**

Let a particle of mass m

_{1}moving with a velocity

**u**

_{1}collides with a particle of mass m

_{2}, initially at rest (i.e.

**u**

_{2}= 0) as shown in Fig. Let us assume that the body of mass m

_{1}is moving along the +x-direction.

After collision with m

_{2}, let it be deflected at an angle Î¸

_{1}with the initial direction with velocity

**v**

_{1}, where

**u**

_{1}and

**v**

_{1}lie in the x-y plane. Let us say that after collision mass m

_{2}is deflected at an angle Î¸

_{2}with the original direction with a velocity

**v**

_{2}.

It is clear that vector

**v**

_{2}will also lie in the x-y plane and will have no component in the z-direction because u

_{1}and

**v**

_{1}have no z-components.

**u**

_{1},

**v**

_{1}and

**v**

_{2}along the x- and y-directions and applying the law of conservation of linear momentum, we have, for the x-components of motion.

and for the y-components of motion

Since the collision is elastic, kinetic energy is also conserved. Therefore,

if we know m

_{1}, m

_{2}and u

_{1}, then there will be four unknown quantities, namely Ï…

_{1}, Ï…

_{2}, Î¸

_{1}and Î¸

_{2}which cannot be determined from these equations.

However, if we know the directions of deflection (Î¸

_{1}and Î¸

_{2}) of the concerned particles, these equations can be solved for velocities Ï…

_{1}and and and Ï…

_{2}.