- #1

- 53

- 0

## Homework Statement

I am basically being asked to use the law of conservation of momentum and kinetic energy to derive two equations for the final velocity of the two masses involved in the system. (i.e. [tex]\acute{v_{1}}[/tex] and [tex]\acute{v_{2}}[/tex])

## Homework Equations

Conservation of Momentum: [tex]m_{1}v_{1}+m_{2}v_{2} = m_{1}\acute{v_{1}}+m_{2}\acute{v_{2}}[/tex] (Simply refered to as Therom 1)

Conservation of Kinetic Energy: [tex]\frac{1}{2}m_{1}v_{1}^{2}+\frac{1}{2}m_{2}v_{2}^{2} = \frac{1}{2}m_{1}\acute{v_{1}}^{2}+\frac{1}{2}m_{2}\acute{v_{2}}^{2}[/tex] (Simply refered to as Therom 2)

For both equations: [tex]m_{1}\neq m_{2} , v_{1}>v_{2}[/tex]

## The Attempt at a Solution

I know that we can get rid of the fractions from Therom 2 by simple factoring and multiplication:

[tex]m_{1}v_{1}^{2}+m_{2}v_{2}^{2} = m_{1}\acute{v_{1}}^{2}+m_{2}\acute{v_{2}}^{2}[/tex]

Then we can solve for an arbitrary final velocity, say [tex]\acute{v_{1}}[/tex], with Theorm 1:

[tex]m_{1}v_{1}+m_{2}v_{2} = m_{1}\acute{v_{1}}+m_{2}\acute{v_{2}}[/tex]

a.[tex]m_{1}\acute{v_{1}} = m_{1}v_{1}+m_{2}v_{2}-m_{2}\acute{v_{2}}[/tex]

b.[tex]\acute{v_{1}} = \frac{m_{1}v_{1}+m_{2}v_{2}-m_{2}\acute{v_{2}}}{m_{1}}[/tex]

c.[tex]\acute{v_{1}} = \frac{m_{1}v_{1}+m_{2}\left(v_{2}-\acute{v_{2}}\right)}{m_{1}}[/tex]

Now we solve for [tex]\acute{v_{2}}[/tex] by substituing in for [tex]\acute{v_{1}}[/tex] the equation that we just solved for:

a.[tex]m_{1}v_{1}^{2}+m_{2}v_{2}^{2} = m_{1}\acute{v_{1}}^{2}+m_{2}\acute{v_{2}}^{2}[/tex]

b.[tex]m_{2}\acute{v_{2}}^{2} = m_{1}v_{1}^{2}+m_{2}v_{2}^{2}-m_{1}\acute{v_{1}}^{2}[/tex]

c.[tex]m_{2}\acute{v_{2}}^{2} = m_{1}v_{1}^{2}+m_{2}v_{2}^{2}-m_{1}\left(\frac{m_{1}v_{1}+m_{2}\left(v_{2}-\acute{v_{2}}\right)}{m_{1}}\right)^{2}[/tex]

(the [tex]\frac{1}{m_{1}^{2}}[/tex] and the [tex]m_{1}[/tex] will cancel out one of the [tex]\frac{1}{m_{1}^{2}}[/tex]'s to get...)

d.[tex]m_{2}\acute{v_{2}}^{2} = m_{1}v_{1}^{2}+m_{2}v_{2}^{2}-\frac{m_{1}^{2}v_{1}^{2}+m_{2}^{2}\left(v_{2}-\acute{v_{2}}\right)^{2}}{m_{1}}[/tex]

Basically its at this point were my brain explodes from all of the m's and v's and their appropriate subscripts. I still have a [tex]\acute{v_{2}}[/tex] on the wrong side of the equals sign and have no idea how to get it on the right side other than voodoo magic and satanic rituals (not really but I'm almost at that point). Any help would be appreciated.