# Thread: Really urgent mechanics question

1. ## Really urgent mechanics question

Hi I am really stuck on the following and have an exam on it in 2 days. Can somebody please help me.

A particle of mass m collides elastically at an angle of 90 degrees with a lighter particle of mass km. Initially both particles had a speed u. After the collision the particles move apart as shown in the attachment show (by the way the angles are both 45 degrees if it is unclear) that k=sqrt(2)-1.

I tried using conservation of linear momentum in horizontal directions giving me 2 equations.

u=(v1^2 ^kv2^2)/sqrt(2)
ku=(v1 -kv2^2)/ sqrt(2)
where v1 is the final velocity of the particle of mass m and v2 is the final velocity of the particle of mass km.

And conservation of kinetic energy tells us that (1+k)u^2=v1^2+v2^2 but I can't put it all together. Please help me

2. Indicating with $\displaystyle \overrightarrow{v_{1}} = v_{1x}\cdot \overrightarrow{i} + v_{1y}\cdot \overrightarrow{j}$ and $\displaystyle \overrightarrow{v_{2}}=v_{2x}\cdot \overrightarrow{i} + v_{2y}\cdot \overrightarrow{j}$ the velocities of the particles after impact, the conservation of momentum and kinetics energy leads to the equations...

$\displaystyle |\overrightarrow{v_{1}}|^{2} + k \cdot |\overrightarrow{v_{2}}|^{2} = (1+k) \cdot u^{2}$

$\displaystyle \overrightarrow{v_{1}} + k \cdot \overrightarrow{v_{2}} = u \cdot (\overrightarrow{i} + k\cdot \overrightarrow{j})$ (1)

... which is equivalent to the scalar equations system...

$\displaystyle v_{1x}^{2} + v_{1y}^{2} + k\cdot v_{2x}^{2} + k\cdot v_{2y}^{2} = (1+k)\cdot u^{2}$

$\displaystyle v_{1x} + k\cdot v_{2x}= u$

$\displaystyle v_{1y} + k\cdot v_{2y}= k\cdot u$ (2)

We have now three equation and four unknown variables, so that we need another equation. This equation is supplied by the supplementar condition that indicates that $\displaystyle \overrightarrow{v_{1}}$ and $\displaystyle \overrightarrow{v_{2}}$ are orthogonal, so that is...

$\displaystyle v_{1x}\cdot v_{2y} = - v_{2x}\cdot v_{1y}$ (3)

... and the problem can be solved...

My question [for experts...] is : the condition (3), necessary to have a well defined problem, isn't in some way arbitrary?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$