# Collisions and loss of kinetic energy

#### Carla1985

##### Active member
Could someone please check my answers so far for this question as its getting very messy so I'm not sure if I've made a mistake :/

Two particles of masses 2m and m are moving along the x−axis with constant velocities u1i and u2i respectively, where u1 > u2. They collide ‘head on’, and the coefficient of restitution of the collision is e. Find the velocities of each of the particles after the collision, and show that the loss in the kinetic energy of the system of two particles is:

$\frac{(1-e^2)m(u_1-u_2)^2}{3}$

so far I have:
the velocities of the particles after impact are v1=v1i and v2=v2i

so 2mu1i-mu2i=2mv1+mv2
ie 2mu1-mu2=2mv1+mv2 (which is my first equation)

then from the law of restitution:
e(u1-u2)=v1-v2 (second equation)

solving the 2 equations I get
v1=1/3((e+2)u1-(e+1)u2)
v2=1/3((2-2e)u1+(2e-1)u2)

the to get the loss of kinetic energy:
(1/2*2m*(v1)2+1/2*m*(v2)2)-(m*(u1)2+1/2*m*(u2)2)

but obviously (v1)2 and (v2)2 gets very messy

Thanks x

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hi Carla1985!

Messy or not, it seems to me you're going the right way.
You did seem to make a sign mistake in your first equation (it should be $2mu_1+mu_2$).
You may want to correct that before getting into the messy stuff.

#### Carla1985

##### Active member
Thats great, I'l keep at it then. Thank you

#### Carla1985

##### Active member
I cannot for the life of me get this into the form i need it in :/

#### Ackbach

##### Indicium Physicus
Staff member
Have you tried working backwards from the form you're supposed to prove? Not an ideal method of solution, but it can sometimes get the job done.

#### Carla1985

##### Active member
thats what I'm attempting atm

#### Carla1985

##### Active member
are the first two equations then:
2mu1-mu2=2mv1+mv2
e(u1-u2)=v1-v2

​as I seem to lose the 1/3 when I do that?

#### Ackbach

##### Indicium Physicus
Staff member
Hmm. So you've got
$$\frac{(1-e^{2})m(u_{1}-u_{2})^{2}}{3} = \frac{\left(1- \left( \frac{v_{1}-v_{2}}{u_{1}-u_{2}} \right)^{2}\right)m(u_{1}-u_{2})^{2}}{3} = \frac{\left(\frac{(u_{1}-u_{2})^{2} - (v_{1}-v_{2})^{2}}{(u_{1}-u_{2})^{2}}\right)m(u_{1}-u_{2})^{2}}{3}$$
$$= \frac{\left((u_{1}-u_{2})^{2} - (v_{1}-v_{2})^{2}\right)m}{3} = \frac{m(u_{1}^{2}- 2u_{1}u_{2}+u_{2}^{2} - v_{1}^{2}+ 2 v_{1}v_{2} - v_{2}^{2})}{3}.$$

I have to run right now. Is this what you get?

#### Carla1985

##### Active member
no, but I went right back and I think I've managed to get the answer thanks for the help x