Elastic 2D Collision with Lots of Unknowns

[mindarson]

Homework Statement

Mass m1, moving to the right, collides with mass m2, which is at rest before the collision. The masses are m1 = 0.1 kg and m2 = 0.2 kg. The scattering angle of m1 will be 30 degrees above the horizontal.

a) Calculate the recoil angle phi of m_2.

b) Calculate the recoil speed of m_2.

Homework Equations

m_1*v_0 = m_2*v_2*cosφ + m_1*v_1*cos30 (mom. cons. in x)

m_2*v_2*sinφ = m_1*v_1*sin30 (mom. cons. in y)

0.5*m_1*v_0^2 = 0.5*m_1*v_1^2 + 0.5*m_2*v2^2 (KE cons.)

The Attempt at a Solution

I'll spare you the messy algebra unless requested. My approach has been to try to eliminate variables, since I have FOUR unknowns (v_0, v_1, v_2, and phi) and only 3 equations. But nothing I've done so far has helped me. Can I really come up with formulas for v_2 and phi in terms of just m_1, m_2, and the 30 deg recoil angle of m_1?

I have also considered trying to use the c.m. frame, but I really don't see how that would help in this problem.

It seems to me the physics here is simple. It's the algebra that's messing me up! Maybe? Can anyone offer a friendly nudge in the right direction?

 

[ehild]

Homework Equations

m_1*v_0 = m_2*v_2*cosφ + m_1*v_1*cos30 (mom. cons. in x)

m_2*v_2*sinφ = m_1*v_1*sin30 (mom. cons. in y)

0.5*m_1*v_0^2 = 0.5*m_1*v_1^2 + 0.5*m_2*v2^2 (KE cons.)

Correct if phi is the angle measured clockwise from the direction of vo.

Can I really come up with formulas for v_2 and phi in terms of just m_1, m_2, and the 30 deg recoil angle of m_1?

No, you can calculate v₂ and phi in terms of v₀.


Arrange all equations so that the quantities labelled by "1" are on one side, and those labelled by "2" on the other side.

Take the square of both momentum equations, and add them: the unknown angle cancels.

Eliminate v₂ using the energy equation. You get a quadratic equation for v1. ...

ehild