Question:

A particle with a rest mass of 3.2 GeV/c^2 decays into two particles, one of which has a rest mass of 0.7 GeV/c^2. If the original particle was at rest, and one of the decay products has a momentum of 2.1 GeV/c, what is the momentum of the other decay product?

Answer:

Given: Rest mass of the original particle (m1) = 3.2 GeV/c^2 Rest mass of one decay product (m2) = 0.7 GeV/c^2 Momentum of one decay product (p2) = 2.1 GeV/c

To find: Momentum of the other decay product (p1)

We can use the conservation of momentum and energy to solve for the momentum of the second decay product. Using the conservation laws, the total initial momentum and energy equals the total final momentum and energy.

The total initial momentum (p_initial) is zero, as the original particle is at rest.

The total final momentum (p_final) can be expressed as: p_final = p1 + p2

Similarly, for the total energy: E_initial = m1*c^2 E_final = E1 + E2

Where c is the speed of light.

The energy-momentum relation for a particle at rest is: E = mc^2

For the second decay product: E2 = m2c^2 p2 = √(E2^2 - (m2c^2)^2)

Using the given values: E2 = (0.7 GeV/c^2)(3.0010^8 m/s)^2 E2 = 0.63 GeV p2 = √((0.63 GeV)^2 - (0.7 GeV)^2) p2 = 0.49 GeV/c

Now, we can solve for the momentum of the first decay product using the conservation equations: p_initial = p_final 0 = p1 + p2 p1 = -p2 p1 = -0.49 GeV/c p1 = 0.49 GeV/c (magnitude)

Therefore, the momentum of the other decay product (p1) is 0.49 GeV/c in the opposite direction of the momentum of the first decay product.