Here it's an elastic collision with no loss of kinetic energy.
We have the conservation of momentum.
m_1u_1+m_2u_2=m_1v_1+m_2v_2m1u1+m2u2=m1v1+m2v2
m_1=5 kgm1=5kg
u_1=12 ms^(-1)u1=12ms−1
m_2=4 kgm2=4kg
u_2=0u2=0
v_1=?v1=?
v_2=?v2=?
5*12+4*0=5*v_1+4*v_25⋅12+4⋅0=5⋅v1+4⋅v2
5v_1+4v_2=605v1+4v2=60............(1)(1)
There is also conservation of kinetic energies
1/2m_1u_1^2+1/2m_2u_2^2=1/2m_1v_1^2+1/2m_2v_2^212m1u21+12m2u22=12m1v21+12m2v22
5*12^2+4*0=5*v_1^2+4*v_2^25⋅122+4⋅0=5⋅v21+4⋅v22
5v_1^2+4v_2^2=7205v21+4v22=720.........(2)(2)
From (1)(1), we get
v_1=(60-4v_2)/5v1=60−4v25
Plugging this value in (2)(2)
5*((60-4v_2)/5)^2+4v_2^2=7205⋅(60−4v25)2+4v22=720
((60-4v_2))^2/5+4v_2^2=720((60−4v2))25+4v22=720
16(15-v_2)^2+20v_2^2=720*5=360016(15−v2)2+20v22=720⋅5=3600
4(15-v_2)^2+5v_2^2=9004(15−v2)2+5v22=900
4(225-30v_2+v^2)+5v_2^2=9004(225−30v2+v2)+5v22=900
900-120v_2+4v_2^2+5v_2^2=900900−120v2+4v22+5v22=900
9v_2^2-120v_2=09v22−120v2=0
v_2(9v_2-120)=0v2(9v2−120)=0
v_2=0v2=0 or v_2=120/9=40/3=13.3ms^-1v2=1209=403=13.3ms−1
v_1=(60-4*40/3)/5=4/3ms^-1v1=60−4⋅4035=43ms−1