A father racing his son has one-third the kinetic energy of the son, who has two-fifths the mass of the father. The father speeds up by 1.5 m/s and then has the same kinetic energy as the son.

Question

What is the Father's initial speed?

What is the Son's initial speed?

Answer 1

Let the father's mass be M sos that the son's mass must be 2M/5. Let the father's initial speed be V, and the son's v.
(1/2)MV^2 = (1/3)(2M/5)v^2
which leads to
V^2 = (4/15)v^2

(1/2)M(V +1.5)^2 = (2M/5) v^2
V^2 + 3V + 2.25 = (4/5) v^2

3V^2 = (4/5)v^2
Now you can eliminate v.

2V^2 -3V -2.25 = 0
That can be solved for a positive V using the quadratic equation. Chack my math.

Answer 2

This answers skips too many steps. It is not helpful.