Final answer:
The speed of the proton with a de Broglie wavelength of 151 pm is calculated using the de Broglie relation that relates wavelength to momentum, and the momentum to the speed by involving the known mass of the proton.
Step-by-step explanation:
To determine the speed of a proton given its de Broglie wavelength, we can use the de Broglie relation which connects the wavelength (λ), and the momentum (p) of a particle. According to de Broglie, λ = h/p, where h is Planck's constant (approximately 6.626 x 10-34 m·kg/s). The momentum of the proton can be related to its velocity (v) using the equation p = mv, where m is the mass of the proton (approximately 1.67 x 10-27 kg).
First, we solve for p using the de Broglie relation: p = h/λ. Substituting the given value of λ (151 pm, which is 151 x 10-12 meters) and converting Planck's constant to MeV·s, we find the momentum in units compatible with the mass of a proton given in MeV/c2. Then, knowing p, we solve for v in p = mv. Since this is a linear accelerator and nonrelativistic velocities are implied, we can ignore relativistic mass increase.
The mass of the proton and the de Broglie wavelength are given, thus allowing us to calculate the speed directly by rearrangement of the equations and substitution of the appropriate constants. Therefore, the velocity of the proton would be found with respect to the de Broglie wavelength. Note that, since 151 pm is very small and the mass of the proton is relatively large, the resulting velocity is expected to be a significant fraction of the speed of light, but still within the nonrelativistic regime as the wavelength value indicates a highly accelerated proton, but not to relativistic speeds.