Elastic Angular Momentum problem

[Violagirl]

Homework Statement

A billiard ball strikes an identical billiard ball initially at rest and is deflected 45 degrees from its original position. Show that if the collision is elastic, the other ball must move at 90 degrees to the first and with the same speed.

Homework Equations

Momentum:
mv = m₁v₁ + m₂v₂

Kinetic energy = 1/2m₁v₁² =1/2m₁v₁² + m₂v₂²

The Attempt at a Solution

I think I've come up with three equations so far in breaking the problem up but I have no idea if I'm the right track or not. Any input is much appreciated.

1. Pf_x=Pi_x
v_xcos 45° + m₂v_xcos θ = v₀

2. Pf_y=Pi_y

v_ysin 45° - v_ysin θ = 0
v_ysin 45°=v_ysin θ

3. 1/2v₀² = v₁²+v₂²

 

[Simon Bridge]

Some observations:
If $v_x$ referrs to the x-component of velocity $v$ then why have you got it paired with trig functions?

If the balls are identical, then there is no need to index mass like that $m_2$ and that mass does not belong in the relation anyway.

If the initial momentum is $m\vec{v}=mv\hat{\imath}$ ...
And the final momentum is $m\vec{v}_1+m\vec{v}_2$ then you want the components in terms of $v_1$ and $v_2$.

I also think you should be doing angles in radiens (get used to it).
Note: $\sin(\pi/4)=\cos(\pi/4)=1/\sqrt{2}$

 

[Violagirl]

Thank you for your input, I'll have to back and reevaluate the equations I found. I set it up the way I did with the hope of eliminating and plugging unknowns into equations. I honestly am confused though on to best go about solving this problem...=/

 

[Simon Bridge]

All conservation problems are worked pretty much the same way - you write a heading "before" and sketch the situation. Then write the equations that describe what you drew.
Do the same for "after". Then state the conservation laws and apply them.

In this case:
before (one mass incoming along the x axis)
$p_x=mv$
$p_y=0$
$K=\frac{1}{2}mv^2$

after (the incoming mass is deflected by θ₁=π/4 from the x axis.[*])
$p_x=v_1\cos(\theta_1)+v_2\cos(\theta_2)$
$p_y=\cdots$
$K=\cdots$

conservation of momentum[\b]
$p_{x,before} = p_{x,after}$
$p_{y,before}\cdots$

conservation of kinetic energy (elastic collision)
$K_{before}\cdots$

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[*] the trig depends on the geometry.
Note: if the first (incoming) ball is deflected by π/4, and the other ball makes an angle of π/2 to the first - then what angle does it make to the x-axis?

 

[haruspex]

v_xcos 45° + m₂v_xcos θ = v₀

Do you mean
v₁cos 45° + v₂cos θ = v₀
where v1 and v2 are the speeds of the balls after collision?