To determine the minimum speed of the moving electron for the collision to be perfectly inelastic, we need to consider the conservation of momentum and energy.

Step 1: Conservation of momentum
In an inelastic collision, the total momentum before and after the collision remains the same. Therefore, we can write the equation:

m_electron * v_electron_initial = (m_electron + m_helium) * v_final

where m_electron is the mass of the electron, v_electron_initial is the initial velocity of the electron, m_helium is the mass of the helium atom, and v_final is the final velocity of both particles after the collision.

Step 2: Conservation of energy
In a perfectly inelastic collision, the kinetic energy is not conserved. Instead, it is converted into other forms of energy, such as heat or potential energy. Therefore, we can write the equation:

(1/2) * m_electron * v_electron_initial^2 = (1/2) * (m_electron + m_helium) * v_final^2

where (1/2) * m_electron * v_electron_initial^2 is the initial kinetic energy of the electron, and (1/2) * (m_electron + m_helium) * v_final^2 is the final kinetic energy of both particles after the collision.

Step 3: Solving the equations
To find the minimum speed of the moving electron, we need to solve the two equations simultaneously. However, since the helium atom is in the ground state, its mass is much larger than the mass of the electron. Therefore, we can neglect the mass of the electron compared to the mass of the helium atom.

m_electron * v_electron_initial = m_helium * v_final

(1/2) * m_electron * v_electron_initial^2 = (1/2) * m_helium * v_final^2

Step 4: Simplifying the equations
Since we neglected the mass of the electron, we can cancel out the mass terms:

v_electron_initial = v_final

v_electron_initial^2 = v_final^2

Step 5: Finding the minimum speed
To find the minimum speed, we need to consider the case where the final velocity is zero. This means that the moving electron comes to a complete stop after the collision. Therefore, we can substitute v_final = 0 into the equations:

v_electron_initial = 0

v_electron_initial^2 = 0

Step 6: Conclusion
From the equations, we can see that the minimum speed of the moving electron for the collision to be perfectly inelastic is zero. This means that the electron needs to come to a complete stop after colliding with the stationary helium atom in the ground state.

Question

To determine the minimum speed of the moving electron for the collision to be perfectly inelastic, we need to consider the conservation of momentum and energy.

Step 1: Conservation of momentum
In an inelastic collision, the total momentum before and after the collision remains the same. Therefore, we can write the equation:

m_electron * v_electron_initial = (m_electron + m_helium) * v_final

where m_electron is the mass of the electron, v_electron_initial is the initial velocity of the electron, m_helium is the mass of the helium atom, and v_final is the final velocity of both particles after the collision.

Step 2: Conservation of energy
In a perfectly inelastic collision, the kinetic energy is not conserved. Instead, it is converted into other forms of energy, such as heat or potential energy. Therefore, we can write the equation:

(1/2) * m_electron * v_electron_initial^2 = (1/2) * (m_electron + m_helium) * v_final^2

where (1/2) * m_electron * v_electron_initial^2 is the initial kinetic energy of the electron, and (1/2) * (m_electron + m_helium) * v_final^2 is the final kinetic energy of both particles after the collision.

Step 3: Solving the equations
To find the minimum speed of the moving electron, we need to solve the two equations simultaneously. However, since the helium atom is in the ground state, its mass is much larger than the mass of the electron. Therefore, we can neglect the mass of the electron compared to the mass of the helium atom.

m_electron * v_electron_initial = m_helium * v_final

(1/2) * m_electron * v_electron_initial^2 = (1/2) * m_helium * v_final^2

Step 4: Simplifying the equations
Since we neglected the mass of the electron, we can cancel out the mass terms:

v_electron_initial = v_final

v_electron_initial^2 = v_final^2

Step 5: Finding the minimum speed
To find the minimum speed, we need to consider the case where the final velocity is zero. This means that the moving electron comes to a complete stop after the collision. Therefore, we can substitute v_final = 0 into the equations:

v_electron_initial = 0

v_electron_initial^2 = 0

Step 6: Conclusion
From the equations, we can see that the minimum speed of the moving electron for the collision to be perfectly inelastic is zero. This means that the electron needs to come to a complete stop after colliding with the stationary helium atom in the ground state.

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