I am having trouble deriving the equation for this physics problem

Question

Microsoft Word - Math Problems.docxGiven 1/2𝑚1𝑣1i^2 + 1/2𝑚2𝑣2i^2 = 1/2𝑚1𝑣1f^2 + 1/2𝑚2𝑣2f^2

and 𝑚1𝑣1i + 𝑚2𝑣2i = 𝑚1𝑣1f + 𝑚2𝑣2f

(1) derive symbolic expressions of 𝑣1f and 𝑣2f as a function of 𝑚1 , 𝑚2, 𝑣1i , and 𝑣2i

Answer 1

From your first equation, I canceled (1/2) from both side and bring initial and final velocities of first mass to the left side, and second mass to the right side.

m1(V1i²) - m1(V1F²) = m2(V2F²) - m2(V2i²)

we know that (x²-y²)= (x-y)(x+y) →

m1[ (V1i - V1F)×(V1i + V1F)] = m2[(V2F - V2i)×(V2F+V2i)] (I)

From your second equation: m1(V1i) + m2(V2i) = m1(V1F) + m2(V2F)

m1(V1i) - m1(V1F) = m2(V2F) - m2(V2i) → m1 [ V1i - V1F ] = m2 [ V2F - V2i ] (II)

Now I substitute m1 [ V1i - V1F ] from equation (II) in equation (I):

m2 [ V2F - V2i ] [(V1i + V1F)] = m2[(V2F - V2i)×(V2F+V2i)] → (V1i+V1F) = (V2F+V2i)

or V1i - V2i = V2F - V1F (III) This is an important formula (difference between velocities before collision is equal with difference between them after collision)

From (III) → V1F = V2F+V2i - V1i

Now I substitute above equation in this equation m1(V1i) + m2(V2i) = m1(V1F) + m2(V2F)

m₁(V1i) + m2(V2i) = m1(V2F+V2i - V1i) + m2(V2F) → 2m1(V1i)+(m2-m1)(V2i) = (m1+m2)(V2F)

V2F = [2m1/(m1+m2)] (V1i) + [(m2-m1)/(m1+m2)] (V2i)

If I substitute above equation V2F in this formula V1i - V2i = V2F - V1F (III) , I will find V1F:

V1F = [(m1-m2)/(m1+m2)] (V1i) + [2m2/(m1+m2)] (V2i)