Question

A ball with a mass of `5 kg` is rolling at `8 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

`v_1^'=24/7" " m/s`
`v_2=80/7" "m/s`

Explanation:

`m_1=5" " kg`
`v_1=8" "m/s`

`m_2=2" " kg`
`v_2=0`

`"momentum before collision:"`
`P_b=m_1*v_1+m_2*v_2`
`P_b=5*8+2*0`
`P_b=40+0`
`P_b=40 " "kg*m/s`

`"momentum after collision:"`
`P_a=m_1*v_1^'+m_2*v_2^'`
`P_a=5*v_1^'+2*v_2^'`

`P_b=P_a" conservation of momentum"`

`40=5*v_1^'+2*v_2^'" (1)"`

`m_1*v_1+m_2*v_2=m_1*v_1^'+m_2*v_2^' " (3)"`
`1/2m_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)'" (4)"`

`"we obtain the equation of "v_1+v_1^'=v_2+v_2^'" ;"`
`"if we arrange together the equation (3) and (4)"`

`8+v_1^'=0+v_2^'`
`v_2=8+v_1^'" "(5)`

`"now; let's use the equation of (1)"`
`40=5*v_1^'+2*(8+v_1^')`
`40=5.v_1^'+16+2*v_1^'`
`40-16=7*v_1^'`
`24=7*v_1^'`
`v_1^'=24/7" " m/s`

`"now;let's use the equation of (5)"`
`v_2=8+24/7`

`v_2=(56+24)/7`

`v_2=80/7" "m/s`

`"is solution true ?"`
`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)'`

`5*8^2+0=5*(24/7)^2+2*(80/7)^2`

`5*64=5*576/49+2*6400/49`

`320=(2880+12800)/49`

`320=15680/49`

`color(green)(320=320" True")`