Question

Derive the expression m_1 u_1+m_2 u_2=m_1 v_1+ m_2 v_2

Answer 1

heay buddy here is your answer

Answer 2

I I find that it is easier to change the frame of reference to the center of momentum frame. In the COM frame, the collision is between two objects carrying equal and opposite momentum. This means that both objects "bounce", simply reversing their velocities. 

So the procedure is:
1. Convert the initial velocities to the center of momentum frame.
2. Negate the resulting velocities.
3. Convert back to the original frame of reference. Step 1 amounts to finding a velocity u to add to each initial velocity so that 

(U1 + u)*m1 = -(U2 + u)*m2

Solve for u: u = -(m1*U1 + m2*U2)/(m1 + m2)

Step 2 says negate the resulting velocities:

v1 = -(U1 + u) and v2 = -(U2 + u) 


Step 3 says convert back to original frame of reference:

V1 = v1 - u and V2 = v2 - u

Putting steps 2 & 3 together:

V1 = -(U1 + 2*u) and V2 = -(U2 + 2u)

I think that if you substitute the value for u from above into these expressions that you'll find the desired derived expressions.