All right, here is the question A father racing his son has 1/3 the kinetic energy of the son, who has 1/3 the mass of the father. The father speeds up by 1.6 m/s and then has the same kinetic energy as the son. So to being with, I set up two equations equal to each other, since the only unknown is initial velocity. I set it up as (2/3)(1/2)(3m)(1.6)squared = (1/2)(3m)(v initial +1.6) squared So to sum this up, I used 3m to represent the fathers weight in reference to the kid. The KE of the first half is multiplied times 2/3 because the increase of 1.6 m/s equalized the KE of the father and the boy, therefore the 1.6 m/s accounted for the other 2/3 of the energy. And on the other side, the v initial plus 1.6 m/s represented the final velocity that equaled the sons KE. I then cancelled out the 3m and the 1/2 and graphed the quadratic equation, but I am getting an answer of .359 m/s, and that is not right. How can i fix this, and am I even starting off correctly?
The increase of 1.6m/s by the father triples his KE, which makes his KE equal to his son's KE. The equation you wrote does not quite express this. Besides, you may not need to involve the son at first. Can you get the father's Vi just from knowing that Vi + 1.6m/s triples his KE compared to just running at Vi? And then you can use masses and KEs to figure out the boy's Vi.