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Chapter 9: Problem 48

A \(^{238} \mathrm{U}\) nucleus is moving in the \(x\) -direction at \(5.0 \times 10^{5} \mathrm{m} / \mathrm{s}\) when it decays into an alpha particle \(\left(^{4} \mathrm{He}\right)\) and \(\mathrm{a}^{234}\) Th nucleus. The alpha moves at \(1.4 \times 10^{7} \mathrm{m} / \mathrm{s}\) at \(22^{\circ}\) above the \(x\) -axis. Find the recoil velocity of the thorium.

Short Answer

Expert verified
The recoil velocity of the thorium and the direction can be obtained through the conservation of momentum before and after the decay of the Uranium. The important calculation required is the separation of the alpha particle velocity into its component vectors and then balancing overall momentum to find the remaining unknown velocity vector of the thorium nucleus.

Step by step solution

01

Analyze the given data

Before decay, we have a Uranium nucleus moving on x-axis with a speed of \(5.0 \times 10^{5} \mathrm{m} / \mathrm{s}\). After it decays, the alpha particle moves at \(1.4 \times 10^{7} \mathrm{m} / \mathrm{s}\) at 22 degrees above x-axis. The task is to find the recoil speed of Thorium nucleus.
02

Calculation of components

The recoil speed and direction of thorium nucleus can be calculated by conserving momentum. First, we must obtain the alpha particle's velocity components. Alpha's velocity in x-direction would be \(v_{ax} = 1.4 \times 10^{7} \mathrm{m} / \mathrm{s} \cdot cos(22^{\circ})\) and in the y-direction would be \(v_{ay} = 1.4 \times 10^{7} \mathrm{m} / \mathrm{s} \cdot sin(22^{\circ})\).
03

Conservation of Momentum in the x-direction

Before decay, the total momentum in the x-direction was due to Uranium nucleus \(p_{ux} = m_u v_u\) and after decay, it is the sum of the momenta of alpha particle and thorium in the x-direction. So, \(m_u v_u = m_a v_{ax}+ m_t v_{tx}\). From this equation, we can solve for the velocity of thorium in x-direction \(v_{tx} = (m_u v_u - m_a v_{ax}) / m_t \)
04

Conservation of Momentum in the y-direction

Before decay, there was no motion in the y-direction, so the total momentum in the y-direction must be zero after decay. It is the sum of the momenta of alpha and thorium nucleus in the y-direction. So, \(0 = m_a v_{ay} - m_t v_{ty}\). From this equation, we can solve for the velocity of thorium in y-direction, \(v_{ty} = m_a v_{ay} / m_t \)
05

Calculation of thorium recoil velocity

The magnitude of the thorium recoil velocity can be obtained using the Pythagorean theorem: \(v_t = \sqrt{{v_{tx}}^2 + {v_{ty}}^2}\)
06

Calculation of the direction of thorium nucleus

Now that we have both the components of velocity, we can calculate the direction of motion of the Thorium nucleus by taking the tangent of velocity components \( \theta = tan^{-1} (v_{ty} / v_{tx})\)

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