The question is about the asymmetric fission of Uranium-235 (^{235}_{92}U), a common process in nuclear reactions. In nuclear fission, a heavy nucleus splits into smaller nuclei along with the emission of neutrons and release of energy. Here, we need to determine where the maximum energy appears during this asymmetric fission process.

The options provided are:

1. Kinetic energy of the emitted neutrons
2. Energy of the emitted gamma rays
3. Kinetic energy of the lighter fission fragments
4. Kinetic energy of the heavier fission fragments

When ^{235}_{92} U undergoes fission, it usually splits into two smaller nuclei, known as fission fragments, along with the emission of 2-3 neutrons and gamma rays. The total energy released in these reactions is roughly 200 MeV.

The distribution of this energy is as follows:

• Kinetic energy of the fission fragments: This is where the majority of the energy in a fission event is invested. It accounts for about 165 MeV of the total energy. This energy is further split between the two fission fragments.
• Kinetic energy of the emitted neutrons: About 5-10 MeV is shared by these neutrons.
• Energy of the emitted gamma rays: Approximately 5-10 MeV of the energy is released as gamma radiation.
• Kinetic energy distribution among fission fragments: Because of momentum conservation, the lighter fragment receives more kinetic energy compared to the heavier fragment. This is due to the inverse mass relationship (lighter object acquires greater velocity).

Therefore, the kinetic energy of the lighter fission fragments is where the maximum energy appears during the fission of ^{235}_{92}U, which is in alignment with the principle of conservation of momentum where the smaller mass fragment would move faster.

Thus, the correct answer is:

Kinetic energy of the lighter fission fragments.