If travelling all same speeds, which of the following matter waves have the shortest wavelength?

To address the question regarding which matter waves have the shortest wavelength when traveling at the same speed, we need to consider the relationship between wavelength, speed, and momentum for matter waves, governed by de Broglie's hypothesis.

1. ### Break Down the Problem

We will look at the relevant formula that links the wavelength of a matter wave to its momentum. According to de Broglie's equation: $\lambda = \frac{h}{p}$ where:

• $\lambda$ is the wavelength,
• $h$ is Planck’s constant ($6.626 \times 10^{-34} \, \text{Js}$),
• $p$ is the momentum of the particle.

2. ### Relevant Concepts

The momentum $p$ of an object can be expressed as: $p = mv$ where:

• $m$ is the mass,
• $v$ is the velocity.

Since all matter waves are traveling at the same speed, the momentum will depend on the mass of the particles.

3. ### Analysis and Detail

1. Effect of Mass on Wavelength:

   • From the relationship $\lambda = \frac{h}{mv}$, we can see that if $v$ is constant (same speed for all particles), then $\lambda$ is inversely proportional to $m$.
   • Therefore, a larger mass $m$ results in a shorter wavelength $\lambda$.

2. Comparison of Matter Waves:

   • If you compare various particles, the matter waves of heavier particles (e.g., protons) will have shorter wavelengths compared to lighter particles (e.g., electrons) at the same speed.

4. ### Verify and Summarize

To summarize, the matter wave with the shortest wavelength, when traveling at the same speed, will be associated with the particle that has the greatest mass.

Final Answer

Among matter waves traveling at the same speed, the one with the greatest mass will have the shortest wavelength. Therefore, if comparing protons, neutrons, electrons, and similar particles, the protons (or other heavier particles) will exhibit the shortest wavelength.