I am facing trouble with the following question
A rod $AC$ of length $l$ and mass $m$ is kept on a horizontal smooth surface. It is free to rotate and move. A particle of same mass $m$ moving on the plane with velocity $v$ strikes rod at $B$ elastically as shown in the figure below.
The question is to find the angular velocity of the rod and its final velocity.
I tried basically by applying conservation of energy , linear momentum and angular momentum.
Let the final situation be something like this
Now conservation of energy gives $$1/2mv^2=1/2m(v_5^2+v_6^2)+1/2m(v_3^2+v_4^2)+1/2(1/12ml^2w^2)$$
and conservation of angular momentum $$m(4v/5)(3l/4)=(1/12ml^2w^ 2)w-mv_4(3l/4)$$
and by linear momentum in two directions $$ 3v/4=v_3+v_6$$ and $$4v/5=v_5-v_4$$
But I have 5 unknowns and I could manage only 4 equations. Moreover solving these equations is lengthy and I think there must be some quicker way to solve this being a entrance exam question with time restrictions. Any help would be highly appreciated. Thanks.