Kinetic energy equation is: Ek = 0.5mv^2
Ek is proportional to v^2
The father's kinetic energy doubles as he speeds up by 1.0 m/s. If his energy doubled then his velocity must have increased by a factor of root 2. So we can deduce the following equation where 'x' is the father's initial velocity.
x • sqrt(2) = x + 1
Rearrange for x:
x(sqrt(2)) - x = 1
x(sqrt(2) - 1) = 1
x = 1/(sqrt(2) - 1) m/s
Using the information you provided, we can also deduce two equations for the son and father in terms of m, v and E. x is the initial velocity of the father and y is the velocity of the son:
0.5E = 0.5 • m • x^2 (father)
E = 0.5 • 0.5m • y^2 (son)
Multiply the father's equation by two and equate the two equations (as both will have E = ...) :
m • x^2 = 0.5 • 0.5m • y^2
Rearrange for y:
x^2 = 0.25y^2 (m cancels out)
y^2 = 4x^2
y = 2x
Therefore y (the speed of the son) is equal to twice the initial velocity of the father which we worked out as 1/(sqrt(2) - 1)
So y = 2/(sqrt(2) - 1) m/s
Which is 4.8 m/s to 2.s.f