m_1=5" " kg
v_1=8" "m/s
m_2=2" " kg
v_2=0
"momentum before collision:"
P_b=m_1*v_1+m_2*v_2
P_b=5*8+2*0
P_b=40+0
P_b=40 " "kg*m/s
"momentum after collision:"
P_a=m_1*v_1^'+m_2*v_2^'
P_a=5*v_1^'+2*v_2^'
P_b=P_a" conservation of momentum"
40=5*v_1^'+2*v_2^'" (1)"
m_1*v_1+m_2*v_2=m_1*v_1^'+m_2*v_2^' " (3)"
1/2m_1*v_1^2+1/2*m_2*v_2^2=1/2*m_1*v_1^('2)+1/2*m_2*v_2^(2)'" (4)"
"we obtain the equation of "v_1+v_1^'=v_2+v_2^'" ;"
"if we arrange together the equation (3) and (4)"
8+v_1^'=0+v_2^'
v_2=8+v_1^'" "(5)
"now; let's use the equation of (1)"
40=5*v_1^'+2*(8+v_1^')
40=5.v_1^'+16+2*v_1^'
40-16=7*v_1^'
24=7*v_1^'
v_1^'=24/7" " m/s
"now;let's use the equation of (5)"
v_2=8+24/7
v_2=(56+24)/7
v_2=80/7" "m/s
"is solution true ?"
cancel(1/2)m_1*v_1^2+cancel(1/2)*m_2*v_2^2=cancel(1/2)*m_1*v_1^('2)+cancel(1/2)*m_2*v_2^(2)'
5*8^2+0=5*(24/7)^2+2*(80/7)^2
5*64=5*576/49+2*6400/49
320=(2880+12800)/49
320=15680/49
color(green)(320=320" True")