Final answer:
The ratio of kinetic energies when a rocket's velocity is suddenly tripled is often misunderstood. If we simply triple and then square the velocity, the ratio of the final kinetic energy to the initial kinetic energy would be 1:81, but the correct ratio after just tripling the velocity is 1:9.
Step-by-step explanation:
The question asks about the ratio of kinetic energies when the velocity of a rocket is suddenly tripled. The kinetic energy (KE) of an object is given by the equation KE = 1/2 m v2, where m is the mass and v is the velocity of the object. Initially, the kinetic energy of the rocket is KE1 = 1/2 m v2. After tripling the velocity, the new kinetic energy is KE2 = 1/2 m (3v)2 = 9/2 m v2 because (3v)2 = 9v2.
Therefore, the ratio of the two kinetic energies is KE2 / KE1 = (9/2 m v2) / (1/2 m v2) = 9. This implies the second kinetic energy is 9 times greater than the first. However, the question likely meant to ask for a comparison between the second and the initial energy before tripling, not between the resultant energies after squaring the tripled speed. The proper comparison of KE before and after tripling the velocity should result in a ratio of 1:9, not 1:12 as stated in the question. However, if we consider the kinetic energy after tripling and then squaring the velocity (which is not the usual interpretation), we get a ratio of 1:81 not 1:12.