Question

A ball with a mass of `2 kg` is rolling at `9 m/s` and elastically collides with a resting ball with a mass of `1 kg`. What are the post-collision velocities of the balls?

Answer 1

No `cancel(v_1=3 m/s)`
No `cancel(v_2=12 m/s)`

the speed after collision of the two objects are see below fro explanation:
`color(red)(v'_1 = 2.64 m/s, v'_2 = 12.72 m/s)`

Explanation:

`"use the conversation of momentum"`
`2*9+0=2*v_1+1*v_2`
`18=2*v_1+v_2`
`9+v_1=0+v_2`
`v_2=9+v_1`
`18=2*v_1+9+v_1`
`18-9=3*v_1`
`9=3*v_1`
`v_1=3 m/s`
`v_2=9+3`
`v_2=12 m/s`

Because there are two unknown I am not sure how you able to solve the above without using, conservation of momentum and conservation of energy (elastic collision). The combination of the two yields 2 equation and 2 unknown which you then solve:

Conservation of "Momentum":
`m_1v_1 + m_2v_2 = m_1v'_1 + m_2v'_2` =======> (1)

Let, `m_1 = 2kg; m_2 = 1 kg; v_1=9m/s; v_2=0m/s`

Conservation of energy (elastic collision):
`1/2m_1v_1^2 + 1/2m_2v_2^2 = 1/2m_1v'_1^2 + 1/2m_2v'_2^2` =======> (2)

We have 2 equations and 2 unknowns:
From (1) ==> `2*9 = 2v'_1 + v'_2; color(blue)(v'_2 = 2(9-v'_1))` ==>(3)
From (2) ==> `9^2 = v'_1^2 + 1/2v'_2^2` ===================> (4)

Insert `(3) => (4)`:

`9^2 = v'_1^2 + 1/2*[color(blue)[2(9-v'_1)]]^2` expand
`9^2 = v'_1^2 + 2(9^2-18v'_1 + v'_1^2)`
`2v'_1^2 -36v'_1 + 9^2 = 0` solve the quadratic equation for `v'_1`
Using the quadratic formula:
`v'_1 = (b +-sqrt(b^2 - 4ac)/2a); v'_1 => (2.64, 15.36)`
The solution that make sense is 2.64 (explain why?)
Insert in (3) and solve `color(blue)(v'_2 = 2(9-color(red)2.64) = 12.72`
So the speed after collision of the two objects are:
`v'_1 = 2.64 m/s, v'_2 = 12.72`

Answer 2

`v_1=3 m/s`
`v_2=12 m/2`

Explanation:

`m_1*v_1+m_2*v_2=m_1*v_1'+m_2*v_2^'" (1)"`
`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) "`
`m_1*v_1^2+m_2*v_2^2=m_1*v_1^('2)+m_2*v_2^('2)" (2)"`
`m_1*v_1-m_1*v_1^'=m_2*v_2^'-m_2*v_2" redeployment of (1)"`
`m_1(v_1-v_1^')=m_2(v_2^'-v_2)" (3)"`
`m_1*v_1^2-m_1*v_1^('2)=m_2*v_2^('2)-m_2*v_2^2" redeployment of (2)"`
`m_1(v_1^2-v_1^('2))=m_2(v_2^('2)-v_2^2)" (4)"`
`"divide :(3)/(4)"`
`(m_1(v_1-v_1^'))/(m_1(v_1^2-v_1^('2)))=(m_2(v_2^'-v_2))/(m_2(v_2^('2)-v_2^2))`
`(v_1-v_1^')/((v_1^2-v_1^('2)))=((v_2^'-v_2))/((v_2^('2)-v_2^2))`
`v_1^2-v_1^('2)=(v_1+v_1^')*(v_1-v_1^') ; v_2^('2)=(v_2^'+v_2)*(v_2^'-v_2)`
`v_1+v_1^'=v_2+v_2^'`