Problem 34 - College Physics

Question

A particle has a rest mass of \(6.64 \times 10^{-27} \mathrm{kg}\) and a momen- tum of \(2.10 \times 10^{-18} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) . (a) What is the total energy (kinetic plus rest energy) of the particle? (b) What is the kinetic energy of the particle? (c) What is the ratio of the kinetic energy to the rest energy of the particle?

Step-by-Step Solution

Answer

(a) Total energy is \(5.99 \times 10^{-10} \ J\), (b) KE is \(2.00 \times 10^{-12} \ J\), (c) Ratio is \(0.00335\)."

Step 1: Understanding the formulas needed

To solve these problems, we need to understand two physics formulas: First, for the total energy of a particle, we use the formula \(E = \sqrt{(pc)^2 + (m_0c^2)^2}\), where \(p\) is momentum, \(c\) is the speed of light, and \(m_0\) is the rest mass. Second, the kinetic energy \(KE\) is found by subtracting the rest energy \(m_0c^2\) from the total energy. Finally, to find the ratio of kinetic to rest energy, use \(\frac{KE}{m_0c^2}\).

Step 2: Calculating Rest Energy

The rest energy \(E_0\) is calculated with the formula \(E_0 = m_0c^2\), where \(c\) is the speed of light \(\approx 3.00 \times 10^8 \ m/s\). Here, \(E_0 = (6.64 \times 10^{-27} \ kg) \cdot (3.00 \times 10^8 \ m/s)^2\). Therefore, \(E_0 = 5.97 \times 10^{-10} \ J\).

Step 3: Calculating Total Energy

Using the total energy formula \(E = \sqrt{(pc)^2 + (m_0c^2)^2}\), substitute \(p = 2.10 \times 10^{-18} \ kg \cdot m/s\) and \(m_0c^2 = 5.97 \times 10^{-10} \ J\). So, \(E = \sqrt{(2.10 \times 10^{-18} \cdot 3.00 \times 10^8)^2 + (5.97 \times 10^{-10})^2}\), which simplifies to \(E \approx 5.99 \times 10^{-10} \ J\).

Step 4: Calculating Kinetic Energy

The kinetic energy \(KE\) is \(KE = E - E_0\). Substitute the values: \(KE = 5.99 \times 10^{-10} \ J - 5.97 \times 10^{-10} \ J= 2.00 \times 10^{-12} \ J\).

Step 5: Calculating Ratio of Kinetic to Rest Energy

The ratio is \(\frac{KE}{E_0}\). Substitute the known values: \(\frac{2.00 \times 10^{-12} \ J}{5.97 \times 10^{-10} \ J} \approx 0.00335\). This means the kinetic energy is \(0.335\%\) of the rest energy.

Key Concepts

Total Energy

Total energy in the context of relativity refers to the sum of an object's rest energy and kinetic energy. It's essential as it demonstrates how energy is conserved and transferred even when objects move at significant fractions of the speed of light. The total energy formula is given by: \[ E = \sqrt{(pc)^2 + (m_0c^2)^2} \] where:

• \( E \) is the total energy
• \( p \) is the momentum
• \( c \) is the speed of light
• \( m_0 \) is the rest mass

This equation combines both the energy a particle has when at rest and the additional energy due to its motion. In our specific exercise, we used this equation to calculate the particle's total energy, resulting in a value of approximately \( 5.99 \times 10^{-10} \, J \). Such calculations are fundamental in modern physics, especially when dealing with fast-moving particles in accelerators.

Rest Energy

Rest energy is the energy that is stored within an object due to its mass. It represents the amount of energy an object has solely based on its mass, independent of its movement. Calculated with Einstein's famous equation:\[ E_0 = m_0c^2 \]where:

• \( E_0 \) is the rest energy
• \( m_0 \) is the rest mass
• \( c \) is the speed of light (\( \approx 3.00 \times 10^8 \, m/s \))

For our particle, the rest energy was found to be \( 5.97 \times 10^{-10} \, J \). Understanding rest energy is pivotal as it lays the groundwork for recognizing that mass can be viewed as a form of energy, which was a revolutionary idea introduced by Einstein.

Kinetic Energy

Kinetic energy is the energy of motion. In relativistic physics, it differs from the classical expression because relativistic speeds factor changes in mass into account. The kinetic energy can be determined from the total energy and rest energy with:\[ KE = E - E_0 \]In this exercise, after calculating the total energy \( E \) and rest energy \( E_0 \), the kinetic energy was determined to be \( 2.00 \times 10^{-12} \, J \). This signifies the additional energy the particle has due to its motion. Kinetic energy is crucial in studying particle physics, where particles often travel at speeds approaching that of light, making classical kinetic energy calculations inadequate.

Momentum

Momentum in physics expresses how much motion a particle has. In relativity, momentum considers changes in mass and energy at high velocities, separate from classical definitions. The formula used is:\[ p = \frac{m_0v}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]where:

• \( p \) is the momentum
• \( m_0 \) is the rest mass
• \( v \) is the velocity
• \( c \) is the speed of light

However, in many cases like our example, the momentum is given, \( p = 2.10 \times 10^{-18} \, kg \cdot m/s \). Even though we know the velocity may be a significant fraction of the speed of light, calculations focus on energy dynamics. In modern physics, accurately understanding momentum is paramount for experiments that explore the fundamental nature of particles.