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## Homework Statement

Two balls with mass m and 4m collide at the location x=y=0 and stick. Their initial velocities just before the collision can be represented as v1=(i+j) v and v2=(j-i)v' respectively. Their final velocity vf makes an angle θ with the +x axis. Find v and v' in terms of vf and θ.

## Homework Equations

p=mv

Thing x-component = Thing * cos θ (i)

Thing y-component = Thing * sin θ (j)

## The Attempt at a Solution

Momentum is conserved meaning that the initial x and y components are equal to the final x and y components of momentum, so:

mv1x-4mv2x=(m+4m)vfcosθ

mv1y+4mv2y=(m+4m)vfsinθ

Simplifying the equations above and removing m yields

v1x-4v2x=5vfcosθ

v1y+4v2y=5vfsinθ

Then we know that the vectors v1 and v2 are equal to (i+j)v and (j-i)v' repsectively. We can represent each vector as

v1 = vi+vj

v2 = -v'i+v'j

where the i represents the x value and j represents the y component.

Now subsituting those components into the equation above yields

vi+4v'i=5vfcosθ

vj+4v'j=5vfcosθ

If I solve for v' first, I get v=(5vfcosθ-4iv')/i if we solve for v in the first equation above, and then subsituting that in to the second equation gets j((5vfcosθ-4iv')/i)+4jv'=5vfsinθ

If we distribute the j and multiply the whole thing by i we get

5vfcosθij-4ijv'+4jiv'=5vfisinθ

However the two 4ijv' cancel and then I can't solve for v'. The same thing happens trying to solve for v.

Why does the v'/v keep canceling out?