MathDude

v20 = 1/4*(v20 + 1)

v20 = 1/4*v20 + 1/4

v20 = 1/3 m/s

1/4*v₁₀² = v₂₀²

1/4*v₁₀² = 1/9

v₁₀² = 4/9

v10 = 2/3 m/s

Son's starting speed = 1/3 m/s

Father's starting speed = 2/3 m/s

I have no idea why the answers are different because I can't follow your logic well, but atleast this part of mine was wrong

(1/2)M(V+1)² = (1/2)mv²  So (V+1)² = v2/2

Instead it has to cancel out the M and m by (1/2)M(V+1)² = (1/2)mv²  So (V+1)² = v²/4

that should account for the discrepancies not the velocity thing I said before

This assumes the father and son have the same start velocity.

A line

log₁₃₆(680) = 1.3276106198925393

K = 1/2*mv²

K₁₀  = 2 * K₂₀

^{K₁₀ = K_2F}

m₁ = 1/2 m₂

K₁₀ = 1/4*m₂v²

K₂₀ = 1/2*m₂*v²

1/4*m₂v₁₀² = 2 * 1/2*m₂v₂₀²

1/4*v₁₀² = v₂₀²

^K_2F = 1/2*m₂*(v₂₀ + 1)²

1/4*m₂*v₁₀² = 1/2*m₂*(v₂₀ + 1)² 

1/4*v₁₀² = 1/2*(v₂₀ + 1)² 

1/4*v₁₀² = v₂₀² 

v₂₀² = 1/2*(v₂₀ + 1)² 

v_{20 =} 1/2*(v₂₀ + 1)

v₂₀ = 1/2_*v₂₀ + 1/2

v_{20 = 1 m/s}

1/4*v₁₀² = v₂₀²

1/4*v₁₀² = 1

v₁₀2 = 4

v₁₀ = 2 m/s

Son's starting speed = 2 m/s

Father's starting speed = 1 m/s