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Chapter 9: Problem 28

A neutron (mass 1 u) strikes a deuteron (mass 2 u), and they combine to form a tritium nucleus. If the neutron's initial velocity was \(28 \hat{\imath}+17 \hat{\jmath}\) Mm/s and if the tritium leaves the reaction with velocity \(12 \hat{\imath}+20 \hat{\jmath} \mathrm{Mm} / \mathrm{s},\) what was the deuteron's velocity?

Short Answer

Expert verified
The initial velocity of the deuteron was \(4\hat{i}+21.5\hat{j} \mathrm{Mm/s}\).

Step by step solution

01

Initial momentum of the neutron

The neutron's momentum before the collision can be calculated using the formula \( \textbf{p}_{\text{initial neutron}} = m_{\text{neutron}} * \textbf{v}_{\text{initial neutron}} \), where \( m_{\text{neutron}} = 1u \) and \( \textbf{v}_{\text{initial neutron}} = 28\hat{i}+17\hat{j} \mathrm{Mm/s} \). Thus, \( \textbf{p}_{\text{initial neutron}} = 1u * (28\hat{i}+17\hat{j} \mathrm{Mm/s}) = 28\hat{i}+17\hat{j} \mathrm{u.Mm/s}. \)
02

Final momentum of the tritium

The tritium's momentum after the collision is given by the formula \( \textbf{p}_{\text{final tritium}} = m_{\text{tritium}} * \textbf{v}_{\text{final tritium}} \), where \( m_{\text{tritium}} = 3u \) and \( \textbf{v}_{\text{final tritium}} = 12\hat{i}+20\hat{j} \mathrm{Mm/s} \). Hence, \( \textbf{p}_{\text{final tritium}} = 3u * (12\hat{i}+20\hat{j} \mathrm{Mm/s}) = 36\hat{i}+60\hat{j} \mathrm{u.Mm/s}. \)
03

Initial momentum of the deuteron

The deuteron's momentum before the collision can now be computed using conservation of momentum, which states that \( \textbf{p}_{\text{initial neutron}} + \textbf{p}_{\text{initial deuteron}} = \textbf{p}_{\text{final tritium}} \). Solving this equation for \( \textbf{p}_{\text{initial deuteron}} \) gives \( \textbf{p}_{\text{initial deuteron}} = \textbf{p}_{\text{final tritium}} - \textbf{p}_{\text{initial neutron}} = (36\hat{i}+60\hat{j} - 28\hat{i}-17\hat{j}) \mathrm{u.Mm/s} = 8\hat{i}+43\hat{j} \mathrm{u.Mm/s}. \)
04

Initial velocity of the deuteron

Finally, calculating the deuteron's initial velocity involves dividing its initial momentum by its mass. Hence, \( \textbf{v}_{\text{initial deuteron}} = \textbf{p}_{\text{initial deuteron}} / m_{\text{deuteron}} = (8\hat{i}+43\hat{j} \mathrm{u.Mm/s}) / 2u = 4\hat{i}+21.5\hat{j} \mathrm{Mm/s}. \)

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