Center of Mass Frame for Elastic Collision

[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!

[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!

Hopefully this may help you

[heartofaragorn]

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

[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]]

Is that right answer?

[cristo]

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

For LaTex on the forum, use [ tex ]...[ /tex ] tags, without the spaces inside the square brackets, instead of $.

[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

[rootX]

Thank you so much!

(My maxima makes it for me :D, but puts $$)

[heartofaragorn]

Wow, thank you for all of the help!