How do I solve elastic collisions?

[Brainiac11]

A 2-kg ball is moving at 3 m/s toward the right. It elastically collides with a 4-kg ball that is initially at rest. Calculate the velocities of the balls after the collision.

I know that kinetic energy is conserved in elastic conditions, but I don't know how to use that to calculate this. I tried solving this using the conservation of momentum, but i ended up with 2(3) + 4(0) = 2vf₁ + 4vf₂ and I can't solve for both variables. I know PE = mgh and KE = (1/2) mv²

 

[HallsofIvy]

Yes, in all collisions, in which there is no external force, momentum is preserved and that is what your equation, 2(3)+ 4(0)= 6= 4vf1+ 2vf2 or, dividing through by 2, 2vf1+ vf2= 3. And, as you say, in an elastic collision, kinetic energy is conserved. Since this doesn't involve change in height, the "potential energy" equation is irrelevant. And, as you say, kinetic energy is "$(1/2)mv^2$" the total kinetic energy before the collision is $(1/2)(2)(9)+ (1/2)(4)(0)= 9$. After the collision, with speeds vf1 and vf2, the kinetic energy is $(1/2)(2)(vf1^2)+ (1/2)(4)(vf2^2)= vf1^2+ 2vf2^2$. Since the kinetic energy does not change, $vf1^2+ 2vf2^2= 9$.

Solve the two equations $2v1+ vf2= 3$ and $vf1^2+ 2vf2^2= 9$ for vf1 and vf2.

 

[Brainiac11]

Thank you so much. I follow all of your work and understand it!