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Chapter 8: Problem 33

Many nuclear collisions studied in laboratories are analyzed in a frame of reference relative to the laboratory. A proton, with a mass of \(1.6605 \cdot 10^{-27} \mathrm{~kg}\) and traveling at a speed of \(70.0 \%\) of the speed of light, \(c\), collides with a tin\(116\left({ }^{116} \mathrm{Sn}\right)\) nucleus with a mass of \(1.9096 \cdot 10^{-25} \mathrm{~kg} .\) What is the speed of the center of mass with respect to the laboratory frame? Answer in terms of \(c\), the speed of light.

Short Answer

Expert verified
Solution: To find the speed of the center of mass with respect to the laboratory frame, we first calculate the relativistic momenta of the proton and the tin nucleus. Then, we compute the total momentum and mass of the system and use these values to determine the velocity of the center of mass. After solving for \(v_{CM}\) in terms of the speed of light, we find the desired result.

Step by step solution

01

Calculate the momentum of the proton

Using the relativistic momentum formula, we can calculate the momentum of the proton. \(p_p = \frac{m_pv_p}{\sqrt{1 - v_p^2/c^2}}\) where \(m_p = 1.6605 \cdot 10^{-27} \mathrm{~kg}\) is the mass of the proton and \(v_p = 0.7c\) is its velocity.
02

Calculate the momentum of the tin nucleus

Since the tin nucleus is stationary in the laboratory frame, its momentum will be zero. \(p_{tin} = 0\)
03

Calculate the total momentum of the system

Now we can find the total momentum of the system by adding the momenta of the proton and the tin nucleus. \(p_{total} = p_p + p_{tin}\)
04

Calculate the total mass of the system

The total mass of the system is the sum of the masses of the proton and the tin nucleus. \(M = m_p + m_{tin}\) where \(m_{tin} = 1.9096 \cdot 10^{-25} \mathrm{~kg}\) is the mass of the tin nucleus.
05

Calculate the velocity of the center of mass

Using the equation for the velocity of the center of mass, we can find the speed of the center of mass with respect to the laboratory frame. \(v_{CM} = \frac{p_{total}}{M}\) Now we just need to plug in the values we found in the previous steps and solve for \(v_{CM}\). After plugging the values in and solving for \(v_{CM}\), we'll find the speed of the center of mass with respect to the laboratory frame in terms of \(c\), the speed of light.

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Most popular questions from this chapter

Suppose you place an old-fashioned hourglass, with sand in the bottom, on a very sensitive analytical balance to determine its mass. You then turn it over (handling it with very clean gloves) and place it back on the balance. You want to predict whether the reading on the balance will be less than, greater than, or the same as before. What do you need to calculate to answer this question? Explain carefully what should be calculated and what the results would imply. You do not need to attempt the calculation.

Find the following center-of-mass information about objects in the Solar System. You can look up the necessary data on the Internet or in the tables in Chapter 12 of this book. Assume spherically symmetrical mass distributions for all objects under consideration. a) Determine the distance from the center of mass of the Earth-Moon system to the geometric center of Earth. b) Determine the distance from the center of mass of the Sun-Jupiter system to the geometric center of the Sun.

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