An alpha particle with kinetic energy 7.70 MeV collides with an 14 N nucleus at rest, and the tw

step 1 So here in the first part which is the kinetic energy of oxygen nucleus, the kinetic energy of o

An alpha particle with kinetic energy 7.70 MeV collides with...

An alpha particle with kinetic energy 7.70 MeV collides with an 14 $\mathrm{N}$ nucleus at rest, and the two transform into an 17 $\mathrm{O}$ nucleus and a proton. The proton is emitted at $90^{\circ}$ to the direction of the incident alpha particle and has a kinetic energy of 4.44 MeV. The masses of the various particles are alpha particle, $4.00260 \mathrm{u} ;$ i4 $\mathrm{N}$ 14.00307 u;proton, 1.007825 $\mathrm{u}$ ;and $17 \mathrm{O}, 16.99914 \mathrm{u} .$ In MeV, what are (a) the kinetic energy of the oxygen nucleus and (b) the $Q$ of the reaction? (Hint: The speeds of the particles are much less than $c . )$

An alpha particle with kinetic energy 7.70 MeV collides with an 14 $\mathrm{N}$ nucleus at rest, and the two transform into an 17 $\mathrm{O}$ nucleus and a proton. The proton is emitted at $90^{\circ}$ to the direction of the incident alpha particle and has a kinetic energy of 4.44 MeV. The masses of the various particles are alpha particle, $4.00260 \mathrm{u} ;$ i4 $\mathrm{N}$ 14.00307 u;proton, 1.007825 $\mathrm{u}$ ;and $17 \mathrm{O}, 16.99914 \mathrm{u} .$ In MeV, what are
(a) the kinetic energy of the oxygen nucleus and (b) the $Q$ of the reaction? (Hint: The speeds of the particles are much less than $c . )$

Key Concepts

Conservation of Momentum

In any collision or nuclear reaction the vector sum of the momenta before the reaction must equal the sum after the reaction. This principle applies even when particles are deflected at angles relative to the original direction, meaning that both the magnitude and direction of momentum must be accounted for in the analysis. For non-relativistic particles, momentum is calculated as the product of mass and velocity, and it is often necessary to use its components along orthogonal axes to set up simultaneous equations that describe the motion.

Conservation of Energy

Energy conservation in nuclear reactions means that the total energy—which includes both kinetic energy and rest mass energy—is conserved. In non-relativistic problems, the kinetic energies of the particles can be added to any energy released or absorbed (the Q-value) of the reaction. This principle allows one to relate the initial kinetic energies of the reacting particles to the kinetic energies of the resulting particles, taking into account any energy change due to mass differences.

Reaction Q-Value

The Q-value of a nuclear reaction is the net amount of energy released or absorbed during the process. It is defined as the difference between the total rest mass energies of the reactants and products. A positive Q-value indicates that energy is released, whereas a negative Q-value indicates that energy is absorbed. Calculating the Q-value involves converting mass differences into energy, typically using the mass-energy equivalence relation in appropriate energy units.

Non-Relativistic Kinematics

When dealing with particles moving at speeds much less than the speed of light, the non-relativistic approximation is used where kinetic energy is given by (1/2)mv². This simplifies the analysis by avoiding the complexities of relativistic mechanics. In this regime, the conservation laws of momentum and energy are employed using classical formulas, which is particularly useful in solving problems involving nuclear reactions and particle collisions.

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