Using conservation of momentum to find final velocity in a collision

elsternj

1. The problem statement, all variables and given/known data
Two cars, both of mass m, collide and stick together. Prior to the collision, one car had been traveling north at speed 2v, while the second was traveling at speed v at an angle [tex]\phi[/tex]south of east (as indicated in the figure). After the collision, the two-car system travels at speed v_final at an angle [tex]\theta[/tex] east of north.

Find the speed v_final of the joined cars after the collision.
Express your answer in terms of v and [tex]\phi[/tex] .




2. Relevant equations

p=mv
p_i=p_f



3. The attempt at a solution
first i tried to break this down in terms of its components

in the x direction:
m₁v_1icos[tex]\phi[/tex] =(m₁+m₂)V_finalsin[tex]\theta[/tex]

in the y direction:
m₁v₁sin[tex]\phi[/tex]+m₂2v = (m1+m2)v_finalcos[tex]\theta[/tex]

Now here is where I am starting to get mixed up. I have both of my components. (They may be wrong so please help me with those equations) Do I just solve both for v_final and then square them and take the square root? Any insight is much appreciated.

xiaoB

"in the x direction:m1v1icos + m2=(m1+m2)Vfinalsin"

I think maybe your equation got problem because direction x for v2 is zero so momentum for m2 also be zero.
If i m wrong please check it out ! thanks !

elsternj

ah yes, i actually omitted that mass when i originally wrote it and then for some reason when looking at my paper i brought it back for some reason.. that was more or less a typo. i am aware that the second mass does not have momentum in the x direction, i will edit that.

xiaoB

If no going wrong my answer maybe like this:

First take it down then square both sides propose is remove the delta ,

(m1v1icos/(m1+m2)V_final)² =sin²
-----1

[(m1v1sin+m22v)/(m1+m2)v_final]² = cos²----2

1+2 :

(m1v1icos/(m1+m2)V_final)²+[(m1v1sin+m22v)/(m1+m2)v_final]²=1

After that take up [(m1+m2)V_final]² shift it to right then remove the square so that my answer .