A ball with a mass of # 5 kg# is rolling at #5 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
Mar 7, 2017

Answer:

#"The answer " vec v_("2_after")=50/7" "m/s" , "vec v_("1_after")=15/7" "m/s#

Explanation:

#"The definition of the momentum is given as " vec P=m*vec v#
#"Where ;"#
#"m is mass of object and v is its velocity"#

#"if object has not a velocity,it will not have any momentum"#
#(vec P=m*0=0)#

#"The momentum of any object is a vectorial quantity,"#
#"it has a magnitude and direction."#

#"The momentum problems is solved by conservation of momentum"#

#"let us find momentums of object for before impact"#
#"........................................................................................."#
#vec P_("1_before")=m_1* vec v_("1_before")#
#"(momentum before impact for a mass of 5 kg)"#
#"plug m=5 kg and v=5 "m/s#

#vec P_("1_before")=5*5=25 " "kg*m/s#

#vec P_("2_before")=m_2* vec v_("2_before")#
#"(momentum before impact for a mass of 2 kg)"#
#"plug m=2 kg and v=0 "m/s#

#vec P_("2_before")=2*0=0 " "kg*m/s#

#"The vectorial sum of the momentums before impact is"#

# color(red)(Sigma vec P_("before"))=vec P_("1_before")+vec P_("2_before")#

#"if collision is centered, the total momentum"#
#"will be "color(red)(Sigma vec P_("before"))=25+0=25 " "kg*m/s#

#"now let us find the momentums of objects for after impact"#
#.........................................................................................#
#vec P_("1_after")=m_1* vec v_("1_after")#
#"(momentum after impact for a mass of 5 kg)"#

#vec P_("1_after")=5*vec v_("1_after") #

#vec P_("2_after")=m_2* vec v_("2_after")#
#"(momentum after impact for a mass of 2 kg)"#

#vec P_("2_after")=2*vec v_("2_after") #

#"The vectorial sum of the momentums after impact is"#

#color(blue)(Sigma vec P_("after"))=vec P_("1_after")+vec P_("2_after")#

#color(blue)(Sigma vec P_("after"))=5*vec v_("1_after") +2*vec v_("2_after") #

#"Let us write the conservation of momentum"#
#.............................................................................#

#color(red)(Sigma vec P_("before"))=color(blue)(Sigma vec P_("after"))#

#25=5*vec v_("1_after") +2*vec v_("2_after")" " "(1)" #

#vec v_("1_before")+ vec v_("1_after")=vec v_("2_before")+vec v_("2_after")" "(2)#

#"note: you can proof the equation (2) using together momentum "##"and kinetic energy conservations equations. "#

#5+vec v_("1_after")=0+vec v_("2_after")" "(3)#

#vec v_("2_after")=5+vec v_("1_after")#

#"now let us use (1)"#

#25=5*vec v_("1_after") +2(5+vec v_("1_after"))#

#25=5*vec v_("1_after") +10+2*vec v_("1_after")#

#25-10=7*vec v_("1_after")#

#15=7*vec v_("1_after")#

#vec v_("1_after")=15/7" "m/s#

#"use (3)"#

#5+15/7=vec v_("2_after")#

#vec v_("2_after")=50/7" "m/s#