Since momentum is conserved, we can set the momentum of the man equal to the momentum of the woman:
p_man = p_woman
where p is the momentum, given by:
p = m * v
where m is the mass of the person, and v is their velocity.
Since the momentum is the same for both, we can write:
m_man * v_man = m_woman * v_woman
The kinetic energy of each person is given by:
K = 1/2 * m * v^2
where K is the kinetic energy, m is the mass, and v is the velocity.
The ratio of the man's kinetic energy to the woman's kinetic energy is:
K_man / K_woman = (1/2 * m_man * v_man^2) / (1/2 * m_woman * v_woman^2)
We can substitute the expression for the velocity in terms of the momentum, and simplify:
K_man / K_woman = (m_man / m_woman) * (v_man / v_woman)^2
K_man / K_woman = (m_man / m_woman) * (m_woman / m_man)^2
K_man / K_woman = m_woman^2 / m_man^2
Substituting the given values, we get:
K_man / K_woman = (400 N)^2 / (700 N)^2
K_man / K_woman = 0.102
Therefore, the ratio of the man's kinetic energy to the woman's kinetic energy is approximately 0.102.