An alpha-particle collides with an oxygen nucleus which is initially at rest. The alpha-particle is scattered at an angle of 67.0 degrees from its initial direction of motion, and the oxygen nucleus recoils at an angle of 52.0 degrees on the other side of this initial direction. What is the ratio,

To find the ratio of the mass of the oxygen nucleus (m_O) to the mass of the alpha-particle (m_α), we can use the conservation of momentum and conservation of energy principles.

Let's denote the initial velocity of the alpha-particle as v_α and the final velocities after the collision as v'_α for the alpha-particle and v'_O for the oxygen nucleus.

1. Conservation of momentum:
In this problem, since the oxygen nucleus is initially at rest, the conservation of momentum equation can be simplified as:
m_α * v_α = m_α * v'_α + m_O * v'_O

2. Conservation of energy:
Before the collision, the total kinetic energy of the system is given by:
(1/2) * m_α * v_α^2

After the collision, the total kinetic energy of the system is given by:
(1/2) * m_α * (v'_α)^2 + (1/2) * m_O * (v'_O)^2

Now, we need to consider the scattering angles given in the problem:
The alpha-particle is scattered at an angle of 67.0 degrees from its initial direction, which means the angle between the initial direction and final direction of the alpha-particle is 67.0 degrees.

The oxygen nucleus, on the other hand, recoils at an angle of 52.0 degrees on the other side of its initial direction. Therefore, the angle between the initial direction and final direction of the oxygen nucleus is 180 - 52 = 128 degrees.

3. Relationship between scattering angles and velocities:
Using the conservation of momentum equations, we can relate the scattering angles to the final velocities:
tan(67) = (m_α * v'_α) / (m_α * v_α)
tan(128) = (m_O * v'_O) / (m_α * v_α)

4. Solving for the ratio of masses:
Dividing the second equation by the first equation, we get:
tan(128) / tan(67) = (m_O * v'_O) / (m_α * v'_α)

It is important to note that both initial velocities, v_α and v'_α, cancel out from the equation, as they are the same.

Finally, rearranging the equation gives us the desired ratio of masses:
m_O / m_α = (tan(128) / tan(67)) * (v'_O / v'_α)

By knowing the values of the scattering angles and velocities, we can calculate this ratio.

An alpha-particle collides with an oxygen nucleus which is initially at rest. The alpha-particle is scattered at an angle of 67.0 degrees from its initial direction of motion, and the oxygen nucleus recoils at an angle of 52.0 degrees on the other side of this initial direction. What is the ratio, for alpha-particle to oxygen nucleus, of the final speeds of these particles? The mass of the oxygen nucleus is four times that of the alpha particle.

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