$$ \begin{aligned} &\text { Calculate the wavelength of } 1000 \mathrm{~kg} \text { rocket moving with a velocity }\\\ &\text { of } 3000 \mathrm{~km} \text { per hour. }\left(h=6.626 \times 10^{-34} \mathrm{Js}\right) \text { . } \end{aligned} $$

Momentum is a fundamental concept in physics, closely related to the motion of objects. It's defined as the product of an object's mass and its velocity and is denoted by the symbol \textbf{p}. In mathematical terms, momentum is given by the equation: \( p = mv \), where \textbf{m} represents mass and \textbf{v} denotes velocity. This concept is crucial when dealing with the motion of objects, as it embodies both the quantity of motion and the direction in which the motion is happening. For the 1000kg rocket in the exercise, the mass is already given, enabling us to calculate momentum when combined with the velocity—after conversion to the appropriate units, of course. Understanding momentum is important since it relates to the de Broglie wavelength, which connects classical physics with quantum mechanics, demonstrating that moving objects exhibit wave properties.

Planck's constant, symbolized by \textbf{h}, is fundamental in the realm of quantum mechanics and can be thought of as the quantum of action. Its value is \( h = 6.626 \times 10^{-34} \text{ Js} \), and it signifies the scale at which quantum effects become observable. Planck's constant relates the energy carried by a photon to its frequency, as in \( E = hf \), but in the context of de Broglie's hypothesis, it forms the bridge between the momentum of a particle and its wavelength. Thus, the constant plays a pivotal role in calculating the de Broglie wavelength, acting as a conversion factor between what can be seen as two 'worlds': the classical physical description of motion and the wave-like behavior of matter at microscopic scales. In this exercise, it's the divisor that will help us find the wavelength of the rocket.

Velocity conversion is an essential process when dealing with physical equations, as it's important to keep units consistent. In this exercise, we're given the rocket's speed in kilometers per hour (km/h), but the de Broglie wavelength formula requires the velocity to be in meters per second (m/s). The conversion from km/h to m/s involves dividing the speed by 3.6, because there are 3600 seconds in an hour and 1000 meters in a kilometer. Hence, the equation \(1 \text{km/h} = \frac{1}{3.6} \text{m/s}\) allows us to adapt the velocity to the correct units for our calculation: \( 3000 \text{ km/h} = 3000 \times \frac{1}{3.6} \text{ m/s} \). After conversion, you can then proceed to use the velocity in the de Broglie equation effectively.

Wave-particle duality is a concept in quantum mechanics that proposes particles can exhibit both wave and particle characteristics. This idea is encapsulated by the de Broglie hypothesis, which suggests that all matter has an associated wavelength, described by the equation \( \( \lambda = \frac{h}{mv} \) \). The de Broglie wavelength represents the wave aspect of the particle and is especially significant for microscopic particles, like electrons. The ability of particles to show wave-like behavior is central to understanding phenomena such as diffraction and quantum tunnelling. In our rocket example, though it's a macroscopic object with a very small wavelength due to its large mass and velocity, the concept still applies, demonstrating the universal nature of wave-particle duality across the scales of the cosmos.