A ball with a mass of # 5 kg# is rolling at #8 m/s# and elastically collides with a resting ball with a mass of #2 kg#. What are the post-collision velocities of the balls?

1 Answer
Apr 8, 2016

Answer:

#v_1^'=24/7" " m/s#
#v_2=80/7" "m/s#

Explanation:

#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")#