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Collision in Two Dimensions

Elastic Collision in Two Dimensions
Let a particle of mass m1 moving with a velocity u1 collides with a particle of mass m2, initially at rest (i.e. u2 = 0) as shown in Fig. Let us assume that the body of mass m1 is moving along the +x-direction.

After collision with m2, let it be deflected at an angle θ 1 with the initial direction with velocity v1, where u1 and v1 lie in the x-y plane. Let us say that after collision mass m2 is deflected at an angle θ2 with the original direction with a velocity v2.

It is clear that vector v2 will also lie in the x-y plane and will have no component in the z-direction because u1 and v1 have no z-components.

Taking the scalar components of u1, v1 and v2 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 m1, m2 and u1, then there will be four unknown quantities, namely υ1, υ2, θ1 and θ2 which cannot be determined from these equations.

However, if we know the directions of deflection (θ1 and θ2) of the concerned particles, these equations can be solved for velocities υ1 and and and υ2.

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