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 of 45 degrees. The neutron's initial speed is 5.9×105 m/s.
Determine the speeds of the two particles after the collision.
To determine the speeds of the two particles after the collision, we can make use of the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of Momentum: In an elastic collision, the total momentum of the system before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
where m1 is the mass of the neutron, m2 is the mass of the helium nucleus, v1i and v2i are the initial speeds of the neutron and helium nucleus respectively, and v1f and v2f are their final speeds.
2. Conservation of Kinetic Energy: In an elastic collision, the total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision. Mathematically, this can be expressed as:
(1/2) * m1 * v1i^2 + (1/2) * m2 * v2i^2 = (1/2) * m1 * v1f^2 + (1/2) * m2 * v2f^2
where m1, m2, v1i, v2i, v1f, and v2f have the same meaning as described above.
By solving these two equations simultaneously, we can determine the final speeds of the particles.
Given:
- m1 (mass of neutron) = 1
- m2 (mass of helium nucleus) = 4
- v1i (initial speed of neutron) = 5.9 × 10^5 m/s
Let's substitute these values into the equations and calculate the final speeds.
Using the equation of conservation of momentum:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
1 * (5.9 × 10^5) + 4 * 0 = 1 * v1f + 4 * v2f
5.9 × 10^5 = v1f + 4v2f
Now, using the equation of conservation of kinetic energy:
(1/2) * m1 * v1i^2 + (1/2) * m2 * v2i^2 = (1/2) * m1 * v1f^2 + (1/2) * m2 * v2f^2
(1/2) * 1 * (5.9 × 10^5)^2 + (1/2) * 4 * 0 = (1/2) * 1 * v1f^2 + (1/2) * 4 * v2f^2
(1/2) * 1 * (5.9 × 10^5)^2 = (1/2) * 1 * v1f^2 + (1/2) * 4 * v2f^2
(1/2) * (5.9 × 10^5)^2 = (1/2) * v1f^2 + 2 * v2f^2
(5.9 × 10^5)^2 = v1f^2 + 4 * v2f^2
Now we have two equations with two unknowns (v1f and v2f). We can solve these equations simultaneously to find the final speeds of the particles.