# Elastic Collision of a rod which can move and rotate with a particle obliquely

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.

Attempt

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.

• What happens to the incoming mass after collision? Is the collision elastic or inelastic? – Farcher Apr 19 '17 at 6:29
• @Farcher sorry I forgot to mention that collision is elastic. Nothing is known about the velocity of incoming mass after collision. – Pink Apr 19 '17 at 7:06
• If the collision is elastic then the incoming mass can impart no linear momentum to the bar along the direction of the long axis of the bar. – Farcher Apr 19 '17 at 7:10
• @Farcher how can we say that? The collision is oblique in nature. – Pink Apr 19 '17 at 7:13
• Which forces causes a change in momentum along the axis of the rod? – Farcher Apr 19 '17 at 7:16

$e=\frac{Velocity of sepration}{Velocity of approach}$
• Use conventional translatory motion approach. e has no relation with $\omega$. – Abhash Jha Apr 19 '17 at 8:01