Question

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?

Answer 1

`"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`