The Proton on Hydrogen collision problem is treated in the time-dependent formalism using a new self-consistent nuclear trajectory model in conjunction with a simple semi-classical approximation. In this method the nuclear trajectory is dependent on the time-evolution of the electronic wavefunction which is described by a basis of H₂⁺ eigenfunctions. The small-energy, large scattering angle region is well described in this way and agreement with available experimental data is obtained. Tile inclusion of the semi-classical approximation and the use of a larger molecular basis than hitherto employed allow these limits to be quite reasonably extended to describe the small angle and moderate energy region also. Results of charge exchange probabilities and differential-scattering cross-sections in the range 150-1000 e.v. (lab. energy of incident proton beam) are presented along with some inelastic calculations on excitation into the Hydrogen 2p±1 and 2S states. It is further shown that the inclusion of the Gerarde states (2Sσg, 3Dσg) in the basis set has a significant effect on the results obtained for collision energies of 700 e.v. and 1Kev. A new numerical method is described which enables very rapid computation of all quantities required for the basis set, and leads to quick and simple integral calculations.