A particle has a velocity that is \(90 . \%\) of the speed of light. If the wavelength of the particle is \(1.5 \times 10^{-15} \mathrm{~m}\), calculate the mass of the particle.

The speed of light in a vacuum is one of the fundamental constants of nature and has significant importance in physics. Represented by the symbol 'c', it is approximately equal to \(3.0 \times 10^{8}\) meters per second. This immense speed is the ultimate speed limit of the universe, according to Einstein's theory of relativity. It means that nothing can travel faster than light in a vacuum.

Understanding the speed of light is crucial when dealing with high-velocity particles in particle physics. For example, when a particle moves at a speed that is a significant fraction of the speed of light, relativistic effects become important. In the given exercise, the particle's velocity is \(90 \%\) of the speed of light, bringing into play concepts of relativistic mechanics. This velocity determines how its de Broglie wavelength relates to its momentum and, ultimately, its mass.

Particle physics is a branch of physics that studies the fundamental constituents of matter and the forces that govern their interactions. It involves understanding particles such as quarks, leptons, bosons, and how they combine to form atoms, molecules, and more complex structures. Particle physics dives deep into the workings of the universe at the smallest scales where quantum mechanics prevails.

The concept of the de Broglie wavelength, which relates a particle's wavelength to its momentum, is a quantum mechanics phenomenon that has been extended to particle physics. Understanding this relationship is vital for comprehending particle behavior at near-light speeds, as illustrated in our exercise. Here, the subatomic particle displays wavelike properties and by knowing its velocity and wavelength, we are able to determine its mass, which is an important characteristic in particle physics experiments and theories.

Planck's constant, denoted by 'h', is a fundamental constant pivotal in quantum mechanics. Its value is approximately \(6.626 \times 10^{-34}\) joule-seconds (Js). Planck’s constant relates the energy of a photon to the frequency of its electromagnetic wave in the famous equation \(E = hv\) where \(E\) is the energy and \(v\) is the frequency.

The concept of Planck's constant is not only restricted to light but extends to all matter with the de Broglie hypothesis. According to this hypothesis, particle matter, like electrons or other subatomic particles, also exhibit wave-like characteristics. The de Broglie wavelength formula, expressed in terms of Planck's constant, allows us to calculate the wavelength of these particles when their momentum is known. In our exercise, Planck's constant is the key to unlocking the mass of the high-velocity particle, as it provides the crucial link between the observed wavelength and the particle's velocity.