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.
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:
Simplifying the equations above and removing m yields
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
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
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?