The COherent Muon to Electron Transition (COMET) experiment will be searching for the charged lepton flavour violating process µ _ + N(Z, A) -+ e_ + N(Z, A) in aluminium with a single event sensitivity of 3 × 10_17. In order to achieve this sensitivity, COmet will utilise a novel pion capture solenoid and a high power proton beam to create the most intense pulsed muon beam in the world. Because these are innovative technologies, it is not known exactly what the muon yield will be and so, in this thesis, simulations of this section of the experiment have been performed using a variety of hadron production models in order to estimate the muon yield and its uncertainty. Another aspect required to achieve such a low sensitivity is that all possible sources of background need to be known and accounted for. One of the less well measured backgrounds is caused by protons emitted after the muon has been captured by a nucleus and so the AlCaP experiment was set up to measure the rate of this process and, in this thesis, an analysis of the data collected in the first AlCaP run is presented. Throughout this thesis, natural units will be used where h = c = 1.