Question

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.

Short answer

The mass of the particle is \(m_0 = \frac{\frac{6.626 \times 10^{-34} J \cdot s}{1.5 \times 10^{-15} m}}{0.9 \times 3 \times 10^8 m/s} \approx 1.63 \times 10^{-28} kg\).

Step-by-step solution

1. Identify the given information

The given information is the velocity of the particle (\(v = 0.9c\)) and the wavelength (\(λ = 1.5 \times 10^{-15}m\)).

2. Calculate the momentum using the de Broglie wavelength formula

Since \(λ = \frac{h}{p}\), we can solve for \(p\) (momentum) by rearranging the formula: \(p = \frac{h}{λ}\) Plug in the values \(h = 6.626 \times 10^{-34} J \cdot s\) (Planck's constant) and \(λ = 1.5 \times 10^{-15} m\): \(p = \frac{6.626 \times 10^{-34} J \cdot s}{1.5 \times 10^{-15} m}\)

3. Calculate the relativistic velocity

Now, we need to calculate the relativistic velocity using the relativistic length contraction formula \(λ_r = \frac{λ_0}{\sqrt{1-v^2/c^2}}\). First, rearrange the formula to solve for \(λ_0\): \(λ_0 = λ_r\sqrt{1-v^2/c^2}\) Since \(v = 0.9c\), we can make the following substitution: \(λ_0 = λ_r\sqrt{1-0.9^2}\) Plug in the given wavelength value, \(λ_r = 1.5 \times 10^{-15} m\): \(λ_0 = (1.5 \times 10^{-15} m)\sqrt{1-0.81}\)

4. Calculate the mass of the particle

Calculate the mass of the particle using the momentum and rest wavelength. The momentum formula is given by \(p = m_0v\), where \(m_0\) is the rest mass of the particle. Rearrange the formula to solve for \(m_0\): \(m_0 = \frac{p}{v}\) Plug in the values for momentum from step 2 and velocity from step 3: \(m_0 = \frac{\frac{6.626 \times 10^{-34} J \cdot s}{1.5 \times 10^{-15} m}}{0.9 \times 3 \times 10^8 m/s}\)

5. Calculate the rest mass of the particle

Now we can calculate the rest mass by plugging in the values obtained in previous steps into the following equation: \(m_0 = \frac{\frac{6.626 \times 10^{-34} J \cdot s}{1.5 \times 10^{-15} m}}{0.9 \times 3 \times 10^8 m/s}\) Calculate the mass to obtain the final answer.

Key concepts

Relativistic Momentum

Relativistic momentum is a concept that comes into play when dealing with particles moving at significant fractions of the speed of light. Unlike classical momentum, which is simply the product of mass and velocity, relativistic momentum accounts for the effects of special relativity. This is important because, as an object's speed approaches the speed of light (), its momentum increases more sharply than classical physics would predict. A key formula to calculate relativistic momentum is \(p = \frac{m_0 v}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(p\) is momentum, \(m_0\) is the rest mass, \(v\) is velocity, and \(c\) is the speed of light.

• Classical momentum formula: \(p = mv\)
• Relativistic momentum accounts for velocity close to \(c\).

The formula shows how momentum increases with speed, requiring significantly more force to continue accelerating the particle as it nears light speed. This nuance is crucial when calculating or interpreting momentum in high-speed contexts, like in the problem where a particle moves at \(0.9c\).

Planck's Constant

Planck's constant \(h\) is a fundamental constant in physics that relates the energy of a photon to its frequency. It plays a critical role in the field of quantum mechanics, particularly in the calculation of de Broglie wavelengths. The de Broglie wavelength formula is \(\lambda = \frac{h}{p}\), where \(\lambda\) is the wavelength, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{J s}\)), and \(p\) is the momentum of the particle.

• Planck's constant is a link between quantum and classical physics.
• The smaller \(h\), the shorter the wavelength for a given momentum.

In the exercise, Planck's constant is used to convert momentum to wavelength, which is a core calculation in determining the physical properties of particles at atomic and subatomic scales. This constant helps bridge the classical and quantum worlds, providing insight into how particles behave as both waves and particles.

Speed of Light

The speed of light, denoted as \(c\), is a universal physical constant essential in various areas of physics, including relativity and electromagnetism. Its value (\(299,792,458 \text{m/s}\)) is considered invariant, meaning it cannot be surpassed by any object with mass.

• Formally expressed as \(c \approx 3 \times 10^8 \text{m/s}\).
• Used extensively in relativity to describe the upper limit of speed for any particle.

In the context of the exercise, the speed of light is used to determine how the particle's relativistic effects, such as its increased momentum and changed mass, are calculated as its speed nears \(0.9c\). This concept is foundational in analyzing how high-speed effects manifest in the physical behavior of particles and setting the stage for relativity-based calculations. Understanding \(c\) allows for exploration into the nature of space, time, and how they interconnect with motion.