Translational Kinetic energy, am i squaring vectors correctly?

[Kibbel]

Homework Statement

m1 = 5 kg, v1 = < 11, -4, 12 > m/s
m2 = 7 kg, v2 = < -14, 6, -6 > m/s
m3 = 5 kg, v3 = < -27, 35, 24 > m/s

What is the total kinetic energy of this system?

Homework Equations

Momentum(total) = Mass(total)*Velocty(of Center of Mass)

Ktranslation = (Mass(total)*(Velocity of Ceneter of mass)^2)/2

The Attempt at a Solution

I followed this as close to the textbook as possible and still two ways i did it i got it wrong.

I started off by finding the momentum of the entire system by doing m1v1+m2v2+m3v3. Then i divided that by the total mass (m1+m2+m3). this gave me the velocity of the center of mass.

One i had that i plugged that directly into the equaltion to find the total kinetic energy of the system. v << c, and there is no Kinetic vibrational or rotational energy, so its just translational.

so i plugged that into the K translational formula, but neither solutions i got worked out.

I know my velocity of center of mass is correct at <-10.47, 11.59, 8.118>

So i did the dot product multiplication of that to itsself, and got 309.85. squared that to get 96007.6, then mutiplied by 17/2, to get 816060 Joules, which is not accurate.

(17/2)(<-10.47,11.59,8.118>)^2 = 816060 Joules.

other solution i tried was finding the magnitude of Velocity of center of mass, and squaring that and multiplying it by 17/2. which ends up working out exactly the same.


can someone tell me what I am doing wrong? I've followed every rule and haven't ommitted everything and I've done the problem several times so i know I am not making a small mistake

 

[vela]

Think about two masses with equal but opposite momenta. The velocity of the center of mass is zero, but the total kinetic energy of the system is obviously not since the masses are moving. You need to calculate the kinetic energy of the masses separately and then add them up.

 

[Kibbel]

thanks vela, i should've looked at this more conceptually instead of just jumping in and plugging in numbers