Using conservation of momentum to find final velocity in a collision

[elsternj]

Homework Statement

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 $$\phi$$south of east (as indicated in the figure). After the collision, the two-car system travels at speed v_final at an angle $$\theta$$ east of north.

Find the speed v_final of the joined cars after the collision.
Express your answer in terms of v and $$\phi$$ .

Homework Equations

p=mv
p_i=p_f

The Attempt at a Solution

first i tried to break this down in terms of its components

in the x direction:
m₁v_1icos$$\phi$$ =(m₁+m₂)V_finalsin$$\theta$$

in the y direction:
m₁v₁sin$$\phi$$+m₂2v = (m1+m2)v_finalcos$$\theta$$

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 .

 

[rwisz]

Yes, you are on the right track. To solve for the final velocity, you will need to use the conservation of momentum equation, which states that the total momentum before the collision is equal to the total momentum after the collision. Therefore, you can set up the following equation:

m1v1icos\phi + m2v2cos\theta = (m1+m2)vfinalcos\theta

You can also set up a similar equation for the y direction:

m1v1sin\phi + m2v2sin\theta = (m1+m2)vfinalsin\theta

Since the cars stick together after the collision, their masses are combined, and you can simplify the equations to:

m1v1icos\phi + m2v2cos\theta = (m1+m2)vfinalcos\theta

m1v1sin\phi + m2v2sin\theta = (m1+m2)vfinalsin\theta

Now, you can solve for vfinal by dividing both equations by (m1+m2) and then taking the square root of both equations. This will give you the final velocity in terms of v and \phi.