Question

Two protons (each with rest mass m=1.67×10^−27 kg) are initially moving at equal speeds in opposite directions. The protons continue to exist after a collision that produces an η0 particle. The rest mass of the η0 is mη=9.75×10^−28 kg.(a) If the two protons and the η0 are all at rest after the collision, find the initial speed of the protons.(b) What is the kinetic energy Ek of each proton?(c) What is the rest energy Er of the η0?

Answer 1

The initial speed of the protons can be found using conservation of energy, equating the combined kinetic energy of the protons to the rest mass energy of the η0 particle. Kinetic energy Ek is then determined from that speed. Lastly, rest energy Er of the η0 is calculated using E = mηc², with mη being the rest mass of the η0.

Step-by-step explanation:

Initial Speed of Protons

To solve for the initial speed of the protons before the collision that produces an η0 particle, we must use the conservation of energy. Since both protons and the η0 particle are at rest after the collision, their collective kinetic energy is converted into the rest mass energy of the η0 particle. We can set up the energy equation as follows:

2(½mv²) = mηc²

Where m is the mass of a proton, v is the initial speed of the protons, and mη is the rest mass of the η0 particle. Since we are given that m = 1.67×10⁻²⁷ kg and mη = 9.75×10⁻²⁸ kg, we can rearrange and solve for v.

Kinetic Energy Ek of Each Proton

After rearranging the equation, the kinetic energy Ek of each proton which is ½mv² can be found using the initial speed of the protons obtained in the previous step.

Rest Energy Er of the η0 Particle

The rest energy Er of the η0 particle can be calculated using the equation E = mηc². Substituting the given rest mass, we find Er in joules (To convert to MeV, we can use the conversion 1 eV = 1.602×10⁻ J).

Answer 2

The initial speed of the protons is 0. The initial kinetic energy of each proton is zero. The rest energy of the η0 particle is 8.78×10⁻¹¹ J.

Step-by-step explanation:

(a) Initial speed of the protons:

Before the collision, the protons have equal speeds but in opposite directions. Let's denote the initial speed of the protons as v. After the collision, the protons and the produced η0 particle are all at rest. Since momentum is conserved, the total initial momentum of the protons must equal zero:

2mv - 0 = 0

2mv = 0

v = 0

Therefore, the initial speed of the protons is 0.

(b) Kinetic energy of each proton:

Since the protons are initially at rest, their initial kinetic energy is zero.

(c) Rest energy of the η0:

The rest energy of the η0 particle can be found using Einstein's famous equation:

E = mc²

Where E is the rest energy, m is the rest mass, and c is the speed of light. Substituting the given values:

E = (9.75×10⁻²⁸ kg) × (3.00×10⁸ m/s)²

E = 8.78×10⁻¹¹ J

Therefore, the rest energy of the η0 particle is 8.78×10⁻¹¹ J.