Question 1195925
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Two billiard balls with {{{highlight(cross(similar))}}} <U>equal</U> masses have velocities of 3.0 m/s (ball 1) and -2.0 m/s (ball 2) 
when they meet in an elastic head‑on collision. What is the final velocity of the first ball (ball 1) after collision?
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<pre>
Since the collision is ELASTIC, you should apply two conservation laws.

First conservation law is the momentum conservation.
Second conservation law is the mechanical energy conservation.


So, let x be the final velocity of the 1-st ball after collision in m/s,
and let y the final velocity of the 2-nd ball after collision.

The masses of the balls are equal (as it is given). Let m be their mass (in kilograms).


The total momentum before collision was  3m - 2m = m (m*kg/s).
After collision, it remains the same, so we can write 
the momentum conservation equation in the form

    mx + my = m.

After reducing/canceling the common factor "m" in both sides, this equation takes the form

    x + y = 1   m/s    (1)


The total mechanical energy before collision was  {{{(m*3^2)/2}}} + {{{(m*(-2)^2)/2}}} = {{{(13m)/2}}}  joules.
After collision, it remains the same, so we can write 
the mechanical energy conservation equation in the form

    {{{(mx^2)/2}}} + {{{(my^2)/2}}} = {{{(13m)/2}}}  joules.

After reducing/canceling the common factor " {{{(m/2)}}} " in both sides, this equation takes the form

    {{{x^2}}} + {{{y^2}}} = 13   (m/s)^2    (2)


Thus you have a system of two equations (1) and (2).


To solve it, express  y = 1-x  from equation (1) and substitute it into equation (2).
You will get then

    x^2 + (1-x)^2 = 13.


Simplify and find x

    x^2 + 1 - 2x + x^2 = 13

    2x^2 - 2x - 12 = 0

    x^2  -  x -  6 = 0

    (x-3)*(x+2) = 0.


It has two solution:  x = 3  and  x = -2.

We should carefully analyze these solutions.


(a)  If x= 3,  then  y= 1-x = -2.

     It means that velocities after collision are the same as before collision:
     it is as if the balls "penetrate" through each other without the collision,
     "not noticing each other".


     Notice that the equations formally ADMIT such a possibility --- but it is CLEARLY 
     not physical solution/situation.


(b)  If x= -2, then y= 1-x = 1 - (-2) = 3.


     Physically, it means that the balls EXCHANGE with their velocities,
     and it is exactly what an observer may expect at elastic collision 
     of two balls of equal masses. 


<U>ANSWER</U>.  After collision, 1st ball's velocity is -2 m/s; 2nd ball's velocity is 3 m/s.
</pre>

Solved and thoroughly explained.