The kinetic energy can be given by #1/2mv^2#
For the lighter body#->#
Mass#=m_1# and Velocity#=v_1#
For the heavier body#->#
Mass#=m_2# and Velocity#=v_2#
We know that #m_1<m_2#
For the Kinetic energies to be equal the velocity of the lighter body must be more than that of the heavier body.
Therefore, #v_1>v_2#
#1/2m_1(v_1)^2=1/2m_2(v_2)^2#
#cancel(1/2)m_1(v_1)^2=cancel(1/2)m_2(v_2)^2#
#m_1v_1^2=m_2v_2^2# #->color(red)A#
We know momentum is #mv#.
#color(Red)1.# Let's divide equation #color(red)A# by #v_1#
#m_1v_1=(m_2v_2^2)/v_1#
Therefore the momentum of the lighter body is #color(blue)((m_2v_2^2)/v_1)#
#color(red)2.# Let's divide equation #color(Red)A# by #v_2#
#(m_1v_1^2)/v_2=m_2v_2#
Therefore the momentum of the heavier body is #color(blue)((m_1v_1^2)/v_2)#
Now let's compare #(m_2v_2^2)/v_1# and #(m_1v_1^2)/v_2#
We know #m_1v_1^2=m_2v_2^2# and #v_1>v_2#
Here as #v_1>v_2# , the value of #(m_2v_2^2)/v_1# will be less compared to #(m_1v_1^2)/v_2# because #v_1# in the denominator increases as compared to #v_2# and thus decreases the overall value of #(m_2v_2^2)/v_1#.
So now we know #(m_2v_2^2)/v_1 < (m_1v_1^2)/v_2#
From this we can conclude that #m_1v_1<m_2v_2#
