Question

A proton (rest mass 1.67 \(\times\) 10\(^{-27}\) kg) has total energy that is 4.00 times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

Short answer

(a) Kinetic energy: 4.50 × 10^-10 J, (b) Momentum: 1.90 × 10^-19 kg m/s, (c) Speed: 2.83 × 10^8 m/s.

Step-by-step solution

Calculate Rest Energy

The rest energy of a proton is given by the formula \( E_0 = m c^2 \), where \( m = 1.67 \times 10^{-27} \) kg is the rest mass of the proton and \( c = 3 \times 10^8 \) m/s is the speed of light. Calculate \( E_0 \).\[ E_0 = 1.67 \times 10^{-27} \times (3 \times 10^8)^2 \approx 1.50 \times 10^{-10} \text{ J} \]

Determine Total Energy

The total energy \( E \) of the proton is given as 4.00 times its rest energy. Thus, \[ E = 4.00 \times E_0 = 4.00 \times 1.50 \times 10^{-10} = 6.00 \times 10^{-10} \text{ J} \]

Calculate Kinetic Energy

The kinetic energy \( K \) is the total energy minus the rest energy.\[ K = E - E_0 = 6.00 \times 10^{-10} - 1.50 \times 10^{-10} = 4.50 \times 10^{-10} \text{ J} \]

Calculate Momentum

The magnitude of the momentum \( p \) can be found using the formula \( E^2 = (pc)^2 + (E_0)^2 \). Using the given total energy and rest energy, solve for \( p \).\[ (6.00 \times 10^{-10})^2 = (p \times 3 \times 10^8)^2 + (1.50 \times 10^{-10})^2 \]\[ p = \frac{\sqrt{(6.00 \times 10^{-10})^2 - (1.50 \times 10^{-10})^2}}{3 \times 10^8} \approx 1.90 \times 10^{-19} \text{ kg m/s} \]

Calculate Speed of Proton

Using the relationship between momentum and kinetic energy for relativistic particles, the speed \( v \) of the proton can be approximated by \( v = \frac{pc}{E} \).\[ v = \frac{1.90 \times 10^{-19} \times 3 \times 10^8}{6.00 \times 10^{-10}} \approx 2.83 \times 10^8 \text{ m/s} \]

Key concepts

Kinetic Energy

Kinetic energy is the energy a particle possesses due to its motion. In relativistic physics, the kinetic energy is not only dependent on velocity but also on the particle's total energy and rest energy. For a moving proton, the kinetic energy \( K \) can be calculated using the equation \( K = E - E_0 \), where \( E \) is the total energy and \( E_0 \) is the rest energy. Here's what you need to remember:

• The rest energy \( E_0 \) is the energy the proton would have if it were at rest, calculated by \( E_0 = m c^2 \).
• Total energy \( E \) includes both rest energy and kinetic energy.
• Kinetic energy gives us insights into how the energy of a moving proton compares to one that's not moving.

Recognizing the difference between rest and kinetic energy is crucial in understanding how relativistic effects influence high-speed particles.

Rest Energy

Rest energy is a concept that encapsulates the intrinsic energy of a particle at rest. For a proton, this can be found using the formula \( E_0 = m c^2 \), which relates to the mass-energy equivalence principle introduced by Einstein. Points to consider:

• The rest mass \( m \) is a constant value specific to the particle, like \( 1.67 \times 10^{-27} \) kg for the proton.
• The speed of light \( c \) is a universally constant value, approximately \( 3 \times 10^8 \) m/s.
• Rest energy gives the amount of energy contained in the mass of a still particle.

This concept is foundational to understanding how particles behave when subjected to forces and gain motion.

Momentum

Momentum in relativistic physics extends beyond the classical concept of \( p = mv \). Here, it involves the relationship between energy and speed, particularly at high velocities close to the speed of light.The key equation is \( E^2 = (pc)^2 + (E_0)^2 \), where \( E \) is the total energy, \( p \) is the momentum, and \( E_0 \) is the rest energy. To find the momentum, solve for \( p \) by rearranging the formula.Consider these important points:

• Momentum becomes significant when analyzing particles moving near the speed of light.
• The relativistic momentum accounts for increases in mass as speed increases.
• Understanding momentum helps in predicting how particles interact and their paths when external forces are applied.

Grasping relativistic momentum is crucial for analyses in high-energy physics.

Speed of Light

The speed of light \( c \) is not only a key factor in calculating rest energy or momentum but also a crucial constant in understanding relativistic effects. For many problems, such as determining the speed of a proton, it serves as an upper limit for velocity.Highlights include:

• The speed of light is approximately \( 3 \times 10^8 \) m/s, and it remains consistent across all reference frames.
• Nothing can travel faster than this speed. So, any computed velocity like \( v = \frac{pc}{E} \) should always be less than \( c \).
• In relativistic calculations, as objects approach this speed, time and space begin to warp, influencing how we compute energy and momentum.

This concept is not just a measure of speed but an integral part of the framework of modern physics.