Understanding the Nuclear Reaction
In nuclear reactions involving Uranium, the emission of an alpha particle results in the transformation of the original nucleus. To find the velocity of the remaining nucleus after the emission of the alpha particle, we can use the principle of conservation of momentum.
Conservation of Momentum Principle
- Momentum before the reaction = Momentum after the reaction
- The momentum of the system must remain constant if no external forces act on it.
Definitions and Given Values
- Let M be the mass of the Uranium nucleus.
- Let m be the mass of the alpha particle (approximately 4 u).
- Let v_alpha be the velocity of the emitted alpha particle = 1.4 x 10^7 m/s.
- Let v_remaining be the velocity of the remaining nucleus.
Calculation of Remaining Nucleus Velocity
1. Initial Momentum:
- Before the emission, the total momentum is simply the momentum of the Uranium nucleus:
- Momentum_initial = M * 0 (since it is at rest).
2. Final Momentum:
- After the emission, the momentum of the system is:
- Momentum_final = (M - m) * v_remaining + m * v_alpha.
3. Setting the Momenta Equal:
- Since the initial momentum is zero, we set the final momentum to zero:
- 0 = (M - m) * v_remaining + m * v_alpha.
4. Solving for v_remaining:
- Rearranging gives:
- v_remaining = -(m * v_alpha) / (M - m).
Conclusion
The velocity of the remaining nucleus will be in the opposite direction to the alpha particle. It is determined by the mass ratio of the alpha particle and the remaining nucleus, ensuring conservation of momentum is maintained. This showcases the fundamental principles governing nuclear reactions.