A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times that of the neutron. The helium nucleus is observed to move off at an angle =45. The neutron's initial speed is 3.8×105 .
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's first calculate the initial and final momenta of the system.
The initial momentum of the system is given by the product of the mass and velocity of the neutron:
Initial momentum = mass of neutron * initial velocity of neutron
The final momentum of the system can be calculated by breaking it into two components: one along the original path of the neutron and another perpendicular to it. Since the helium nucleus moves off at an angle of 45 degrees, we can find the final momenta along these two directions separately.
The momentum along the original path of the neutron is given by the product of the mass and velocity of the neutron:
Momentum along original path = mass of neutron * final velocity of neutron
The momentum perpendicular to the original path is given by the product of the mass and the final velocity component of the helium nucleus:
Momentum perpendicular to original path = mass of helium nucleus * final velocity component
The magnitude of the final velocity component can be found using the angle given. Since the angle between the original path of the neutron and the final velocity of the helium nucleus is 45 degrees, we can use trigonometry to find the magnitude:
Magnitude of final velocity component = final velocity of helium nucleus * sin(45)
Now, using the principle of conservation of momentum, the total initial momentum of the system equals the combined final momentum along the original path and perpendicular to it:
Initial momentum = Momentum along original path + Momentum perpendicular to original path
Solving this equation for the final velocity of the neutron, we get:
Final velocity of neutron = (Initial momentum - Momentum perpendicular to original path) / mass of neutron
Since this is an elastic collision, the total kinetic energy of the system is conserved.
The initial kinetic energy of the system is given by:
Initial kinetic energy = (1/2) * mass of neutron * (initial velocity of neutron)^2
The final kinetic energy of the system can be calculated by adding the kinetic energies of the final velocities of the neutron and the helium nucleus:
Final kinetic energy = (1/2) * mass of neutron * (final velocity of neutron)^2 + (1/2) * mass of helium nucleus * (final velocity of helium nucleus)^2
Using the principle of conservation of kinetic energy, the initial kinetic energy of the system equals the final kinetic energy:
Initial kinetic energy = Final kinetic energy
Substituting the expressions for the initial and final kinetic energies, we can solve for (final velocity of neutron)^2.
Once we have calculated the final velocity of the neutron and the magnitude of the final velocity component, we can find the final velocity of the helium nucleus using the Pythagorean theorem:
(final velocity of helium nucleus)^2 = (final velocity of neutron)^2 + (final velocity component)^2
Finally, substitute all the values given in the problem statement (mass of neutron, initial velocity of neutron, angle of 45 degrees) to find the final velocities of the neutron and the helium nucleus.