are v ± u. The dot product (v + u) • (v − u) = v2 − u2 = 0, indicating that the vectors are orthogonal. Relativistic analysis Consider the elastic collision scenario in Fig. 6-3 between a moving particle colliding with an equal mass stationary particle. Unlike the Newtonian case, the angle between the two particles after collision is less than 90°, is dependent on the angle of scattering, and becomes smaller and smaller as the velocity of the incident particle approaches the speed of light: The relativistic momentum and total relativistic energy of a particle are given by Conservation of momentum dictates that the sum of the momenta of the incoming particle and the stationary particle (which initially has momentum = 0) equals the sum of the momenta of the emergent particles: Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc2) equals the sum of the total energies of the emergent particles: Breaking down () into its components, replacing with the dimensionless , and factoring out common terms from () and () yields the following: From these we obtain the following relationships: For the symmetrical case in which and () takes on the simpler form: How far can you travel from the Earth? Since nothing can travel faster than light, one might conclude that a human can never travel farther from Earth than ~100 light years. You would easily think that a traveler would never be able to reach more than the few solar systems which exist within the limit of 100 light years from Earth. However, because of time dilation, a hypothetical spaceship can travel thousands of light years during a passenger's lifetime. If a spaceship could be built that accelerates at a constant 1g, it will, after one year, be travelling at almost the speed of light as seen from Earth. This is described by: where v(t) is the velocity at a time t, a is the acceleration of the spaceship and t is the coordinate time as measured by people on Earth. Therefore, after one year of accelerating at 9.81 m/s2, the spaceship will be travelling at v = 0.712c and 0.946c after three years, relative to Earth. After three years of this acceleration, with the spaceship achieving a velocity of 94.6% of the speed of light relative to Earth, time dilation