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 rebound at an angle θ'2 = 44° from the neutron's initial direction. The neutron's initial speed is 7.0 105 m/s. Determine the angle at which the neutron rebounds, θ'1, measured from its initial direction.
What is the speed of the neutron after the collision?
What is the speed of the helium nucleus after the collision?
You will have to write and solve three conservation equations (x momentum, y momentum, and kinetic energy) to solve for the three unknowns (2 final speeds and exit angle). It is often easiest to do this in a coordinate system that moves with the center of mass and then translate back to laboratory coordinates. This would take about a page of calculations and I do not have time to do it for you.
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's start by calculating the angle at which the neutron rebounds, θ'1, measured from its initial direction.
1. Conservation of momentum:
Since the helium nucleus is initially at rest, the total momentum before the collision is equal to the momentum after the collision. The total momentum before the collision is given by the momentum of the neutron:
(mass of neutron) * (initial speed of neutron) = (mass of neutron) * (final speed of neutron) + (mass of helium nucleus) * (final speed of helium nucleus)
(1) * (7.0 * 10^5 m/s) = (1) * (final speed of neutron) + (4) * (final speed of helium nucleus)
2. Conservation of kinetic energy:
An elastic collision conserves kinetic energy. The total initial kinetic energy is equal to the total final kinetic energy. The total initial kinetic energy is given by the kinetic energy of the neutron:
(1/2) * (mass of neutron) * (initial speed of neutron)^2 = (1/2) * (mass of neutron) * (final speed of neutron)^2 + (1/2) * (mass of helium nucleus) * (final speed of helium nucleus)^2
(1/2) * (1) * (7.0 * 10^5 m/s)^2 = (1/2) * (1) * (final speed of neutron)^2 + (1/2) * (4) * (final speed of helium nucleus)^2
3. Solving for the final speed of the neutron:
From equation (1), we can solve for the final speed of the neutron:
(7.0 * 10^5 m/s) = (final speed of neutron) + 4 * (final speed of helium nucleus)
4. Solving for the final speed of the helium nucleus:
From equation (2), we can solve for the final speed of the helium nucleus:
(1/2) * (7.0 * 10^5 m/s)^2 = (1/2) * (final speed of neutron)^2 + (1/2) * (4) * (final speed of helium nucleus)^2
Solving these equations will give us the final answers.
To solve this problem, we can use the principles of conservation of linear momentum and conservation of kinetic energy.
1. To find the angle at which the neutron rebounds, θ'1, measured from its initial direction, we can use the law of conservation of linear momentum. Since the helium nucleus is initially at rest, the total initial momentum is equal to the momentum after the collision.
The initial momentum of the system is given by:
P_initial = m_neutron * v_initial_neutron
The final momentum of the system can be split into two components, one for the neutron and one for the helium nucleus. Considering the x and y components separately, we have:
P_final_x = m_neutron * v_final_neutron * cos(θ'1)
P_final_y = m_neutron * v_final_neutron * sin(θ'1) + m_helium * v_final_helium * sin(θ'2)
Since the total initial momentum is equal to the total final momentum, we can write:
m_neutron * v_initial_neutron = m_neutron * v_final_neutron * cos(θ'1)
0 = m_neutron * v_final_neutron * sin(θ'1) + m_helium * v_final_helium * sin(θ'2)
We can rearrange the second equation to solve for θ'1 in terms of v_final_neutron and v_final_helium:
θ'1 = arcsin((-m_helium * v_final_helium * sin(θ'2)) / (m_neutron * v_final_neutron))
2. To find the speed of the neutron after the collision, we can use the conservation of kinetic energy. Since it is an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The initial kinetic energy is given by:
KE_initial = 0.5 * m_neutron * v_initial_neutron^2
The final kinetic energy can be split into two components, one for the neutron and one for the helium nucleus:
KE_final = 0.5 * m_neutron * v_final_neutron^2 + 0.5 * m_helium * v_final_helium^2
Setting the initial and final kinetic energy equal to each other, we can solve for v_final_neutron:
0.5 * m_neutron * v_initial_neutron^2 = 0.5 * m_neutron * v_final_neutron^2 + 0.5 * m_helium * v_final_helium^2
Rearranging the equation and solving for v_final_neutron:
v_final_neutron = sqrt((m_neutron * v_initial_neutron^2 - 0.5 * m_helium * v_final_helium^2) / m_neutron)
3. To find the speed of the helium nucleus after the collision, we can use the same principle of conservation of kinetic energy. Using the equation derived above for v_final_neutron, we can substitute it into the equation for KE_final and solve for v_final_helium:
v_final_helium = sqrt((2 * (KE_initial - KE_final)) / m_helium)
By substituting the known values into these equations, you can find the answers to both questions.