1) Two balls ($A$ and $B$) of masses $m_a$ and $m_b$ collide in an **oblique** way in a table. Initially, $A$ goes towards $B$ with an initial velocity $V_o$, while $B$ stays at rest. After the inelastic collision $A$ has a velocity $\frac{V_o}{2}$ and moves with a right angle (regards the initial direction of the movement).

The exercise didn't provide any draw; the image shown is what I interpreted from the statement.

a) Determine the angle formed by the ball $B$ and the horizontal after the collision.

b) Determine the velocity of $B$ after the collision.

c) Compare and study the difference between kinetic energy before and after the collision (assess change in kinetic energy) and deduce the rate ma/mb so as to have an inelastic collision.

These are my solutions

As there are not external forces we can assure momentum is conserved. Therefore, we know the following equations are valid:

$m_aV_o, _a + m_bV_o, _b = m_aV_f, _a + m_bV_f, _b$

a) In the x axis case:

$m_aV_o, _a = m_bV_f, _bcos(\alpha)$ ---> **EQ 1**

In the y axis case:

$m_a\frac{V_o}{2} - m_bV_f, _bsin(\alpha) = 0$ ---> **EQ2**

With these two EQs we can get $tg(\alpha)=1/2$; therefore $\alpha=26.56º$

b) Using the two previous EQs I get $V_f, _b= \frac{m_aV_o}{m_bcos(\alpha)}=\frac{m_aV_o}{2sin(\alpha)m_b}$ but I am pretty sure there is something I am missing, as I do not obtain a clear result for $V_f, _b$. What I mean is that I am supposed to obtain a numeric outcome, but as I just acquired the angle as a result, I do not see how I could combine both **EQ1** and **EQ2** so as to obtain a number.

c) First of all I regarded the both masses as point ones. As we have an inelastic collision, there is a loss in kinetic energy. I analysed it as follows:

$KE_o= 1/2m_aV_o^2$ ---> **EQ3**

$KE_f= 1/2m_a(\frac{V_o}{2})^2 + 1/2m_bV_f^2$ ---> **EQ4**

$KE_f < KE_o$ ---> **EQ5**

From these EQs I got:

$\frac{ma}{mb}>(\frac{V_fcos(\alpha)}{V_o})^2 AND \frac{ma}{mb}<(\frac{2V_fsin(\alpha)}{V_o})^2$

But again I got a result slightly conclusive, what makes me doubtful. The reason why I am unsure about my result is that I am supposed to get a numeric outcome in the rate $\frac{m_a}{m_b}\$. I could obtain it if I had a numeric result from b). Notice I am using EQ3, EQ4 and EQ5 at c). As far as I am concerned, my setback here is not physics but algebra. Please let me know if it is backwards.

I am looking for an accurate solution (as much as possible).

Thanks