Question

Suppose the \(_{2}^{4} \mathrm{He}\) nucleus emitted during an alpha decay and the \(\mathrm{e}^{-}\) emitted during a beta decay have the same kinetic energy. Which of the 2 particles is moving at the higher speed? F. The \(_{2}^{4} \mathrm{He}\) nucleus, because it is more massive than the \(e^{-}\) . G. The \(_{2}^{4} \mathrm{He}\) nucleus, because it is less massive than the \(e^{-}\) . H. The \(\mathrm{e}^{-}\) , because it is more massive than the \(_{2}^{4} \mathrm{He}\) nucleus. J. The \(e^{-}\) , because it is less massive than the \(_{2}^{4} \mathrm{He}\) nucleus.

Short answer

Answer: The \(\mathrm{e}^{-}\) (electron) is moving at a higher speed because it is less massive than the \(_{2}^{4} \mathrm{He}\) nucleus.

Step-by-step solution

Understand the kinetic energy formula

The kinetic energy formula is given by \(KE = \frac{1}{2}mv^2\), where m is the mass of the particle and v is its velocity.

Calculate the speed for the \(_{2}^{4} \mathrm{He}\) nucleus

We know that both particles have the same kinetic energy. Let's call the kinetic energy shared by both particles KE. For the \(_{2}^{4} \mathrm{He}\) nucleus, we can rearrange the kinetic energy formula to solve for its speed: \(v_{He} = \sqrt{\frac{2 * KE}{m_{He}}}\).

Calculate the speed for the \(\mathrm{e}^{-}\)

Similarly, we can rearrange the kinetic energy formula to solve for the speed of the \(\mathrm{e}^{-}\) : \(v_{e} = \sqrt{\frac{2 * KE}{m_{e}}}\).

Compare the speeds

To determine which particle is moving at a higher speed, we can compare the expressions for \(v_{He}\) and \(v_{e}\). Since both expressions share the same numerator (\(2 * KE\)), we can compare them by comparing their denominators. Since the \(_{2}^{4} \mathrm{He}\) nucleus is more massive than the \(\mathrm{e}^{-}\) \((m_{He}> m_{e})\), the expression \(\frac{2 * KE}{m_{He}}< \frac{2 * KE}{m_{e}}\).

Conclusion

As a result, we find that the speed of the \(\mathrm{e}^{-}\) is greater than the \(_{2}^{4} \mathrm{He}\) nucleus because the electron is less massive than the helium nucleus. Therefore, the correct answer is J. The \(e^{-}\) , because it is less massive than the \(_{2}^{4} \mathrm{He}\) nucleus.

Key concepts

Kinetic Energy Formula

Kinetic energy is a fundamental concept in physics, describing the energy of motion possessed by an object. The formula for kinetic energy (KE) is given by
\( KE = \frac{1}{2}mv^2 \)
where \( m \) is the mass of the object and \( v \) is its velocity.
This equation tells us that kinetic energy is directly proportional to the mass and the square of the velocity of the object. Hence, when two different particles have the same kinetic energy, their velocities vary inversely with the square root of their respective masses. It's essential to grasp the kinetic energy concept fully, as it plays a pivotal role in solving many types of physics problems, including those related to particle speeds during nuclear decay processes such as alpha and beta decays.

Physics Problem Solving

Problem solving in physics often involves a combination of conceptual understanding and mathematical application. To tackle a physics problem, you should firstly identify the relevant concepts, such as kinetic energy when dealing with motion. Then, consider the appropriate formulas, and finally do the calculations to arrive at a solution.
For instance, in the original problem, we apply the kinetic energy formula to find the velocities of two particles given equal kinetic energies. The steps involve rearranging the formula to solve for velocity, performing some algebraic manipulation, and making a comparison based on the mass of the particles.
These problem-solving techniques are crucial for students to develop, as they provide the tools necessary to break down complex problems into manageable parts, and subsequently apply the appropriate equations to find a solution.

Comparing Particle Speeds

Comparing particle speeds, especially in the context of nuclear decay processes, requires an understanding of kinetic energy and mass. Given that two particles have the same kinetic energy, the particle with the smaller mass will naturally move faster because kinetic energy is shared between mass and velocity according to the kinetic energy formula.
This underpins the relationship between an object's mass and its speed when analyzing motion under the same kinetic energy conditions. It also signifies why, in the exercise, an \(e^{-}\), with far less mass than the \( _{2}^{4} \mathrm{He} \) nucleus, moves at a higher speed despite both having the same kinetic energy.
This conclusion helps reinforce concepts of mass-energy equivalence and the behavior of particles at a subatomic level, illustrating fundamental principles of physics in action.