"Before"
"................................................................"
m_1:3" "kg" ""'mass of the first object'"
vec v_1:12" "m/s" ""'velocity of the first object'"
vec P_1:" 'the first object's momentum before collision'"
vec P_1=m_1*vec v_1
vec P_1=3*12=36 " "kg*m/s
m_2:9""kg" ""'mass of the second object'"
vec v_2=0
vec P_2" 'the second object's momentum before collision'"
vec P_2=m_2*vec v_2
vecP_2=9*0=0
Sigma P_b:" 'The vectorial sum of the momentums before collision'"
vec Sigma P_b=vec P_1+vec P_2
vec Sigma P_b=36+0=36" "kg*m/s
"After"
......................................................................"
vec v_1^':" 'the first object's velocity after collision'"
vec P_1^':" 'the first object's momentum after collision'"
vec P_1^'=m_1*v_1^'=3*v_1^'
vec v_2^':" 'the second object's velocity after collision"
vec P_2^':"' the second object's momentum after collision'"
vec P_2^'=m_2*v_2^'=9*v_2^'
Sigma P_b:" 'The vectorial sum of the momentums after collision'"
vec Sigma P_a=P_1^'+P_2^'
vec Sigma P_a⁼3v_1^'+9v_2^'
"momentum is reserved in elastic collisions."
vec Sigma P_b=vec Sigma P_a
36=3v_1^'+9v_2^'" (1)"
m_1*v_1+m_2*v_2=m_1*v_1^'+m_2*v_2^'" (2)"
1/2*m_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)" (3)"
"you can obtain the equation " v_1+v_1^'=v_2+v_2^'" using (2) and (3)"
"Thus;"
12+v_1^'=0+v_2^'" (4)"
color(red)(v_2^'=12+v_1^')
"plug into (1)"
36=3v_1^'+9(color(red)(12+v_1^'))
36=3v_1^'+108+9v_1^'
12*vec v_1^'=36-108
12*vec v_1^'=-72
vec v_1^'=-72/12
vec v_1^'=-6" "m/s
"now use (4)"
vec v_2^'=12-6
vec v_2^'=6" "m/s
"Testing momentum..."
"Total momentum before:"36 " "kg*m/s
"Total momentum after:"3*(-6)+9*6=-18+54=36" "kg*m/s
Sigma vec P_b=Sigma vec P_a
"Momentum has conserved"
"Testing the kinetic energy..."
E_b=1/2*3*12^2+0=72*3=216J" 'Before'"
E_a=1/2*3*(-6)^2+1/2*9*6^2
E_a=54+162
E_a=216J
E_b=E_a
"The Kinetic energy has conserved"
