# Center of Mass Frame for Elastic Collision

1. Jul 20, 2007

### heartofaragorn

1. The problem statement, all variables and given/known data
Two spheres of equal radius and masses 5.00 kg and 15.0 kg are sliding towards each other along the same straight line across a frictionless, horizontal surface. Vi1 = 15.0 m/s and Vi2= -9.00 m/s, respectively. During the collision, 25% of the total initial kinetic energy is lost...use the results above to find the final velocities of the two objects in the center of mass reference frame.

2. Relevant equations
Kf = 1/2 m1 v1f ^2 + 1/2 m2 v2f ^2
Kf = 1/2 M Vcm ^2 + 1/2 m1 u1f ^2 + 1/2 m2 u2f^2
m1 u1f + m2 u2f = 0

3. The attempt at a solution
This was a multi-step problem in which my professor gave us the above equations. I already solved for Vcm, which is -3.0 m/s. This would give sphere 1 a ui of -12.0 m/s and sphere 2 a ui of 12.0 m/s. I calculated Ki to be 1170 Joules, and calculated the value of Kf to be 75% of the value, or 877.5 Joules. However, I am unsure as to how to use the above equations in order to solve for uf and vf, since there is a change in kinetic energy. I tried several things, like plugging in random numbers, solving for one of the variables and plugging it back in, and have so far been stumped. Please help!

2. Jul 20, 2007

### rootX

you have these things:

Kf >> v1_f and v2_f
pf >> v1_f and v2_f

pf = pi
Kf = 0.75Ki
so two unknowns and two equations!

Last edited: Jul 20, 2007
3. Jul 20, 2007

### heartofaragorn

To what is pi and pf referring? Is that momentum? I'm afraid I'm still a bit lost.

4. Jul 21, 2007

### rootX

YEs

$$0.5\,\left( 5\,{vI}^{2}+15\,{vF}^{2}\right) =877.5$$
$$5\,vI+15\,vF=-60$$
$$[[vI=-\frac{3\,\sqrt{105}+6}{2},vF=\frac{\sqrt{3}\,\sqrt{5}\,\sqrt{7}-6}{2}],[vI=\frac{3\,\sqrt{105}-6}{2},vF=-\frac{\sqrt{3}\,\sqrt{5}\,\sqrt{7}+6}{2}]]$$

oops, here are two equations and their solution using maxima
0.5*(5*vI^2+15*vF^2)=877.5;
5*vI+15*vF=-60
[[vI=-(3*sqrt(105)+6)/2,vF=(sqrt(3)*sqrt(5)*sqrt(7)-6)/2],[vI=(3*sqrt(105)-6)/2,vF=-(sqrt(3)*sqrt(5)*sqrt(7)+6)/2]]

5. Jul 21, 2007

### cristo

Staff Emeritus
For LaTex on the forum, use [ tex ]...[ /tex ] tags, without the spaces inside the square brackets, instead of $. 6. Jul 21, 2007 ### rootX $$0.5\,\left( 5\,{vI}^{2}+15\,{vF}^{2}\right) =877.5$$ $$5\,vI+15\,vF=-60$$ $$[[vI=-\frac{3\,\sqrt{105}+6}{2},vF=\frac{\sqrt{3}\,\sqrt {5}\,\sqrt{7}-6}{2}],[vI=\frac{3\,\sqrt{105}-6}{2},vF=-\frac{\sqrt{3}\,\sqrt{5}\,\sqrt{7}+6}{2}]]$$ Let's see 7. Jul 21, 2007 ### rootX Thank you so much! (My maxima makes it for me :D, but puts$\$)

8. Jul 22, 2007

### heartofaragorn

Wow, thank you for all of the help!