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?
To solve this problem, we can apply the principles of conservation of momentum and conservation of kinetic energy.
1. First, let's consider the conservation of momentum. Since the helium nucleus is at rest initially, the momentum before the collision is only due to the neutron:
Momentum before collision = Momentum after collision
M_n * V_n (initial) = M_n * V_n' (final) + M_He * V_He (final)
where:
M_n = mass of the neutron
V_n (initial) = initial speed of the neutron
V_n' (final) = final speed of the neutron after the collision
M_He = mass of the helium nucleus
V_He (final) = final speed of the helium nucleus after the collision
2. Next, let's consider the conservation of kinetic energy. Since the collision is elastic, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision:
Kinetic energy before collision = Kinetic energy after collision
(1/2) * M_n * (V_n (initial))^2 = (1/2) * M_n * (V_n' (final))^2 + (1/2) * M_He * (V_He (final))^2
3. We are given the following information:
- M_He = 4 * M_n (mass of the helium nucleus is four times that of the neutron)
- V_n (initial) = 7.0 * 10^5 m/s
- θ'2 = 44° (angle at which the helium nucleus rebounds)
4. To find the angle at which the neutron rebounds, θ'1, we can use basic geometry. Since the helium nucleus rebounds at an angle of θ'2 = 44° from the neutron's initial direction, the neutron will rebound at an angle of θ'1 = 180° - 44° = 136° from its initial direction.
5. To find the speeds of the neutron and helium nucleus after the collision, we can solve the two equations from step 1 and step 2 simultaneously using the given information.
Substituting M_He = 4 * M_n into the momentum equation:
M_n * V_n (initial) = M_n * V_n' (final) + 4 * M_n * V_He (final)
Simplifying the equation:
V_n (initial) = V_n' (final) + 4 * V_He (final)
We can substitute this expression for V_n' (final) into the kinetic energy equation from step 2. After substituting and simplifying, we get:
(1/2) * V_n (initial)^2 = (1/2) * (V_n (initial) - 4 * V_He (final))^2 + (1/2) * 4 * V_He (final)^2
Solving these equations will give us the final speed of the neutron and the final speed of the helium nucleus after the collision.
Note: The exact calculations should be done using the given values.