Ch 11: Impulse and Momentum

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 11: Impulse and Momentum

Problem 68

Chapter 11, Problem 68

The nucleus of the polonium isotope ²¹⁴Po (mass 214 u) is radioactive and decays by emitting an alpha particle (a helium nucleus with mass 4 u). Laboratory experiments measure the speed of the alpha particle to be 1.92×10⁷ m/s . Assuming the polonium nucleus was initially at rest, what is the recoil speed of the nucleus that remains after the decay?

Verified step by step guidance

Step 1: Identify the principle of conservation of momentum. Since the polonium nucleus is initially at rest, the total momentum before decay is zero. After the decay, the momentum of the alpha particle and the recoil momentum of the remaining nucleus must sum to zero.

Step 2: Write the equation for conservation of momentum. Let the mass of the alpha particle be \( m_{\alpha} = 4 \, \text{u} \), the mass of the remaining nucleus be \( m_{\text{recoil}} = 214 - 4 = 210 \, \text{u} \), the speed of the alpha particle be \( v_{\alpha} = 1.92 \times 10^7 \, \text{m/s} \), and the recoil speed of the remaining nucleus be \( v_{\text{recoil}} \). The momentum conservation equation is: \( m_{\alpha} v_{\alpha} + m_{\text{recoil}} v_{\text{recoil}} = 0 \).

Step 3: Rearrange the equation to solve for the recoil speed \( v_{\text{recoil}} \). From the conservation of momentum equation, \( v_{\text{recoil}} = - \frac{m_{\alpha} v_{\alpha}}{m_{\text{recoil}}} \). The negative sign indicates that the recoil direction is opposite to the direction of the alpha particle.

Step 4: Convert the masses from atomic mass units (u) to kilograms if needed. The conversion factor is \( 1 \, \text{u} = 1.66 \times 10^{-27} \, \text{kg} \). However, since the ratio of masses is dimensionless, you can directly use the values in atomic mass units for this calculation.

Step 5: Substitute the given values into the formula \( v_{\text{recoil}} = - \frac{m_{\alpha} v_{\alpha}}{m_{\text{recoil}}} \). Use \( m_{\alpha} = 4 \), \( m_{\text{recoil}} = 210 \), and \( v_{\alpha} = 1.92 \times 10^7 \, \text{m/s} \) to calculate the recoil speed. Ensure the units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event. In this case, the initial momentum of the polonium nucleus is zero since it is at rest. After the decay, the momentum of the emitted alpha particle and the recoiling polonium nucleus must balance each other, allowing us to calculate the recoil speed.

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, which consists of two protons and two neutrons (essentially a helium nucleus). This process reduces the mass and atomic number of the original nucleus, resulting in a new element. Understanding alpha decay is crucial for analyzing the changes in mass and momentum during the decay of polonium-214.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In this scenario, the kinetic energy of the emitted alpha particle can be related to the recoil speed of the remaining nucleus. While the question primarily focuses on momentum, recognizing the kinetic energy involved helps in understanding the dynamics of the decay process.

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