A neutron with a mass of 1.67x10^-27 kg traveling east with a kinetic energy of 2.00x10^-21J collides perfectly elastically with helium nucleus with a mass 6.68x10^-27 kg that is initially at rest. After the collision, the neutron has a kinetic energy 1.80x10^-21J. What angle was the neutron deflected during the collision?
To find the angle at which the neutron was deflected during the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's break down the problem step by step:
Step 1: Calculate the initial velocity of the neutron and the helium nucleus.
We can use the formula for kinetic energy to find the initial velocity of the neutron:
K_i = (1/2) * m_n * v_n_i^2
Rearrange the formula to solve for the initial velocity:
v_n_i = √(2 * K_n_i / m_n)
Substituting the given values:
v_n_i = √(2 * 2.00x10^-21 J / 1.67x10^-27 kg)
Step 2: Calculate the final velocity of the neutron and the helium nucleus.
Since the collision is perfectly elastic, we know that the total momentum and total kinetic energy of the system are conserved.
Using the conservation of momentum, we can write:
m_n * v_n_i = m_n * v_n_f + m_He * v_He_f
Since the helium nucleus is initially at rest (v_He_i = 0) and the collision is head-on, we can also write:
m_n * v_n_i = m_n * v_n_f - m_He * v_He_f
Solving these equations simultaneously, we can find the final velocities of the neutron and the helium nucleus.
Step 3: Calculate the angle at which the neutron is deflected.
To find the angle, we need to compare the components of the final neutron velocity in the x and y directions.
v_n_f_x = v_n_f * cos(θ)
v_n_f_y = v_n_f * sin(θ)
Since the velocity of the helium nucleus is zero after the collision, its x and y components are also zero.
Using the conservation of kinetic energy, we can write:
(1/2) * m_n * v_n_i^2 = (1/2) * m_n * v_n_f^2 + (1/2) * m_He * v_He_f^2
Substituting the values and solving the equation will give us the final neutron velocity, v_n_f.
Finally, we can use the components of v_n_f to find the angle θ by using the equation:
θ = tan^(-1)(v_n_f_y / v_n_f_x)
By following these steps and performing the calculations, we can determine the angle at which the neutron was deflected during the collision.