A particle accelerator collides beams of protons and anti-protons with equal and opposite momenta.The aim of an experiment is to create a possible new massive particle with neutral charge and a mass 6 times that of a single proton.Calculate the necessary speed of the protons (and anti-protons)in the collision as a % of the speed of light.
A)85.9% c
B)99.9% c
C)98.7% c
D)94.3% c
E)None of these is correct.

Question

Quizplus · Accepted Answer

Let's assume the velocity of protons and anti-protons as v. Since their momenta are equal and opposite, their net momentum is zero. Therefore, the total energy of the system will be equal to the rest energy of the new particle.

The rest energy of a particle is given by E = mc^2, where m is the mass of the particle and c is the speed of light.

We are given that the mass of the new particle is 6 times that of a single proton. Therefore,

E = 6m_p c^2

where m_p is the mass of a single proton.

In a head-on collision between two particles with momentum p, the total energy of the system is given by

E = sqrt((pc)^2+(mc^2)^2)

where m is the mass of each particle and c is the speed of light.

Since the protons and anti-protons have equal and opposite momenta, the total momentum of the system is zero. Therefore, the total energy of the system is simply equal to 2mc^2.

2mc^2 = sqrt((pc)^2+(mc^2)^2)

Squaring both sides,

4m^2c^4 = p^2 c^2 + m^2c^4

Simplifying,

p = sqrt(3m^2c^2)

Now, substituting the value of m and simplifying,

p = sqrt(3) m_p c

v = p / (gamma * m_p)

where gamma is the Lorentz factor given by gamma = 1/sqrt(1-v^2/c^2)

Substituting the value of p and m_p, gamma can be expressed as

gamma = sqrt(1 + 3/36) = 1.1667

Therefore, v = p / (gamma * m_p) = sqrt(3) m_p c / (1.1667 * m_p) = 0.943 c

Thus, the required velocity of the particles is 94.3% of the speed of light. Therefore, the correct answer is (D).

A Particle Accelerator Collides Beams of Protons and Anti-Protons with Equal