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

"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