Two-Dimensional Elastic Collision

[johnhuntsman]

A particle with speed v1 = 2.64 × 10⁶ m/s makes a glancing elastic collision with another particle that is at rest. Both particles have the same mass. After the collision, the struck particle moves off at 45º to v₁. The speed of the struck particle after the collision is approximately...

The answer is 1.9E6 m/s.

I drew a diagram of the scenario. I know that the two final vectors form a right triangle with the initial v₁. And I have these equations written out:

v_1i = v_1f + v_2f (since momentum is conserved; the mass can be divided out)

v_1i² = v_1f² + v_2f² (since KE is conserved and 0.5m can be divided out)

The first equation broken up into components:

v_1ix = v_1fx + v_2fx

0 m/s = v_1fy + v_2fy

What do I do? Maybe I'm the world's least intuitive man, but I don't see how this system of equations can be solved. All I need is someone to tell me where to start and I'll probably be good to go.

[Edit] Am I supposed to put the components in terms of sine and cosine and go from there? [Edit]

 

[frogjg2003]

Write the energy equation in terms of the components.
You now have 3 equations and 4 unknowns (the final velocities, v_1x, v_1y, v_2x, v_2y).
There is one more equation you forgot:
v_2x = v_2y
which comes from the fact that v₂ is at 45 degrees to v_i.
Now you have 4 equations and 4 unknowns. Have at it.

 

[johnhuntsman]

Write the energy equation in terms of the components.
You now have 3 equations and 4 unknowns (the final velocities, v_1x, v_1y, v_2x, v_2y).
There is one more equation you forgot:
v_2x = v_2y
which comes from the fact that v₂ is at 45 degrees to v_i.
Now you have 4 equations and 4 unknowns. Have at it.

Can you do that? I thought energy didn't have components since it's scalar not vector.

 

[frogjg2003]

It doesn't, it's still one equation. You just write it in terms of the individual components instead of the total velocity.
$$v_1^2=v_{1x}^2+v_{1y}^2\\ v_2^2=v_{1x}^2+v_{2y}^2$$

Another approach would have been to replace every v_2y and v_2x with v₂/√2.

 

[johnhuntsman]

Alright thanks. I got it. I appreciate it : D