Velocity vectors in different directions for momentum

[jb007]

Homework Statement

I'm stuck on this problem, and I don't really know how to approach it.

Homework Equations

Pretty much just p=mv
And the conservation of linear momentum: total initial mv = total final mv

The Attempt at a Solution

I tried just plugging in the variables into the conservation of momentum equation, but it doesn't work out. I know you can't just add velocity vectors that are in different directions, right? They have to have the same I hat or j hat? How would you solve for them?

 

[Orodruin]

I tried just plugging in the variables into the conservation of momentum equation, but it doesn't work out. I know you can't just add velocity vectors that are in different directions, right? They have to have the same I hat or j hat? How would you solve for them?

What do you get when you try conservation of momentum? Can you show us your working? Vectors do add, but they add component by component, for example:
$$
(A\hat i + B \hat j) + (C\hat i + D\hat j) = (A+C)\hat i + (B+D)\hat j.
$$

 

[jb007]

So I used the conservation of p equation like this:

mv₀i + 2m0.5v₀j = mv_f + 2m0.25v₀i

But I know this must be wrong because the vectors here aren't adding by components.

 

[BvU]

It's easier if you follow Oro's notation:$$
m\; (v_0 \hat\imath + 0 \hat\jmath) + 2 m\; (0 \hat\imath + {\textstyle 1\over 2} v_0 \hat\jmath ) = ...$$This gives you two equations: one where you group all the $\hat\imath$ together -- this is the eqauation for conservation of momentum in the x direction -- and one where you group all the $\hat\jmath$ together

From two equations you can solve for two unknowns: the $\hat\imath$ component gives you the velocity component in the x-direction and the $\hat\jmath$ idem y-direction. Together they are the velocity vector, with two components (one or both may be zero, of course).