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Momentum is a fundamental concept in physics. It's often described as the "quantity of motion" an object has. It depends on two factors:

• Mass: Larger objects have more momentum.
• Velocity: Faster objects also have more momentum.

Mathematically, momentum (denoted as "p") is found by the formula: \[ p = m \cdot v \] Where:

• \( m \) is the mass of the object.
• \( v \) is the velocity of the object.

Hence, if either the mass or the velocity increases, the momentum will increase too. Understanding momentum is crucial for solving various physics problems, including those involving de Broglie wavelengths.

Planck鈥檚 constant is a pivotal part of quantum mechanics. It is a tiny number with a symbol \( h \) and a value of approximately \( 6.626 \times 10^{-34} \) m虏 kg / s.
Planck's constant is used to relate the energy of a particle to its frequency. In connection with de Broglie wavelength, it helps to determine the wavelength associated with the momentum of a particle. The de Broglie equation given by: \[ \lambda = \frac{h}{p} \] unites Planck's constant with the particle's momentum \( p \) to find the wavelength \( \lambda \).
This means that smaller particles like electrons have significant wavelengths and wave-like behaviors, while macroscopic objects, like baseballs, have wavelengths that are practically undetectable.

Alpha particles are a type of ionizing radiation ejected by certain radioactive decay processes. They are composed of:

• 2 protons
• 2 neutrons

This composition makes alpha particles the same as helium nuclei. They are relatively massive compared to other particles, which affects their momentum and hence their de Broglie wavelength.
Due to their larger mass, when alpha particles travel at the same velocity as smaller particles, such as protons or electrons, their de Broglie wavelength will be much shorter because the momentum \( (m \cdot v) \) is larger.
Understanding alpha particles is key when dealing with nuclear physics and radiation, providing insights into the behavior of matter at the atomic and subatomic levels.

Protons are one of the fundamental particles that make up matter. They are found in the nucleus of an atom and have:

• a positive charge
• a mass of approximately \( 1.67 \times 10^{-27} \) kg

Because protons are much less massive than alpha particles, they exhibit longer de Broglie wavelengths at the same velocity. This property becomes important in particle physics and quantum mechanics.
In the scenario where protons and alpha particles have the same velocity, as outlined in the exercise solution, the smaller mass of protons means their momentum is less than that of alpha particles.
This results in a longer de Broglie wavelength for protons, highlighting the influence that mass and velocity have on the wave-particle duality that is central to quantum mechanics.