Let #p = m_1 v_1 = m_2 v_2# be the momentum attached to car and lorry.
The work needed to attain rest is in each case
The momentum #m v# has associated a mechanical energy given by #1/2mv^2#. Also #f# is the resistive force the same for both cars and #w# is the dissipated mechanical energy, as breaking work.
#1/2m_1v_1^2=w_1 = f d_1# and
#1/2m_2v_2^2=w_1 = f d_2#
Here #d_1,d_2# are the distances covered until rest. Dividing term to term
#(m_1v_1^2)/(m_2v_2^2)=d_1/d_2#
but #m_1 v_1 = m_2 v_2# so #d_1/d_2=1# hence #d_1=d_2#
