A neutron is an electrically neutral subatomic particle with a mass just slightly greater than that of a proton. A free neutron is radioactive and decays after a few minutes into other subatomic particles. In one experiment, a neutron at rest was observed to decay into a proton (mass 1.67×10^-27 kg) and an electron (mass 9.11×10^-31 kg) . The proton and electron were shot out back-to-back. The proton speed was measured to be 1.0 ×10⁵ m/s, and the electron speed was 3.0×10⁷ m/s. No other decay products were detected. How much momentum did this neutrino 'carry away' with it?

Verified step by step guidance

1. Step 1: Recall the principle of conservation of momentum. In this problem, the total momentum before the decay (initial momentum) must equal the total momentum after the decay (final momentum). Since the neutron is initially at rest, its initial momentum is zero.
2. Step 2: Write the equation for conservation of momentum. Let the momentum of the proton be \( p_p \), the momentum of the electron be \( p_e \), and the momentum of the neutrino be \( p_\nu \). The equation becomes: \( 0 = p_p + p_e + p_\nu \). Rearrange to solve for the neutrino's momentum: \( p_\nu = -(p_p + p_e) \).
3. Step 3: Calculate the momentum of the proton. Momentum is given by \( p = mv \), where \( m \) is the mass and \( v \) is the velocity. For the proton, \( p_p = m_p v_p \), where \( m_p = 1.67 \times 10^{-27} \, \text{kg} \) and \( v_p = 1.0 \times 10^5 \; \text{m/s} \).
4. Step 4: Calculate the momentum of the electron. Similarly, \( p_e = m_e v_e \), where \( m_e = 9.11 \times 10^{-31} \, \text{kg} \) and \( v_e = 3.0 \times 10^7 \; \text{m/s} \).
5. Step 5: Substitute the calculated values of \( p_p \) and \( p_e \) into the equation \( p_\nu = -(p_p + p_e) \) to find the momentum carried away by the neutrino. Ensure the correct signs are used, as the proton and electron are moving in opposite directions.

Key Concepts

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity. It is conserved in isolated systems, meaning the total momentum before an event must equal the total momentum after the event. In particle decay, the momentum of the decay products must equal the momentum of the original particle, allowing us to calculate the momentum carried away by any undetected particles.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum remains constant if no external forces act on it. In the context of the neutron decay, the momentum before the decay (the neutron at rest) must equal the total momentum of the decay products (the proton and electron) plus any momentum carried away by the neutrino, which is not directly observed.

Neutrino Properties

Neutrinos are nearly massless, electrically neutral particles that are produced in certain types of particle decays, such as beta decay. Although they interact very weakly with matter, they carry away energy and momentum during these processes. In this scenario, the neutrino's momentum can be inferred from the momentum of the proton and electron, allowing us to determine how much momentum it carried away during the neutron's decay.