Final answer:
The velocity ratio of a β particle to an α particle, when both have the same kinetic energy, is determined by the square root of the inverse ratio of their masses. Since the α particle is much heavier, the β particle will be faster, leading to a velocity ratio where the β particle is significantly faster. The correct option is D.
Step-by-step explanation:
The question asks about the velocity ratio of a β particle (beta particle) to an α particle (alpha particle) when they possess the same non-relativistic kinetic energy. To find the ratio, we can use the formula for kinetic energy, KE = (1/2)mv², where m is the mass of the particle and v is its velocity.
Given that the kinetic energy (KE) is the same for both particles, we can set up the equation (1/2)m₁v₁² = (1/2)m₂v₂², where subscript 1 refers to the α particle and subscript 2 refers to the β particle.
Because the masses of an α particle and a β particle (which is an electron or positron) are very different (an α particle is much heavier), the velocities must compensate for this difference to have equal kinetic energies. After canceling out the common factors and rearranging the equation, we find that v₂ / v₁ = (√(m₁ / m₂)).
Using the known masses, the mass of an α particle is about 4 times that of a proton, and since a β particle is roughly equivalent to the mass of an electron (which is significantly less than a proton), the β particle will have a much higher velocity. In other words, given the same kinetic energy, the β particle will be much faster than the α particle.
This means that answer choices (a) 1:1, (b) 1:2, and (c) 1:3 are incorrect, leading to option (d) 1:4 as the best representation of the velocity ratio, with the β particle being much faster as it has so much less mass.