Problem 45 - University Physics with Modern Physics

Question

Find the speed of a particle whose relativistic kinetic energy is 50\(\%\) greater than the Newtonian value for the same speed.

Step-by-Step Solution

Answer

The speed is \( v = c \sqrt{\frac{3}{7}} \).

Step 1: Understand the Problem

The problem asks us to find the speed of a particle such that its relativistic kinetic energy is 50% greater than its non-relativistic (or Newtonian) kinetic energy.

Step 2: Relativistic and Newtonian Kinetic Energy Comparison

The relativistic kinetic energy is given by \( K_r = \left( \gamma - 1 \right) mc^2 \) and the Newtonian kinetic energy is \( K_n = \frac{1}{2} mv^2 \), where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \). We are given that \( K_r = 1.5 K_n \).

Step 3: Relate the Energies with the Given Condition

Set up the equation based on the given condition: \( \left( \gamma - 1 \right) mc^2 = 1.5 \times \frac{1}{2} mv^2 \). Simplify the equation to get \( \gamma - 1 = \frac{0.75 v^2}{c^2} \).

Step 4: Substitute Gamma and Solve for v

Substitute the expression for \( \gamma \): \( \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} - 1 = \frac{0.75 v^2}{c^2} \). Multiply throughout by \( \sqrt{1 - \frac{v^2}{c^2}} \) to eliminate the square root, then solve the resulting equation to find \( v \).

Step 5: Algebraic Manipulation

After multiplying through, isolate \( v \) to get \( v = c \sqrt{\frac{3}{7}} \) by solving \( 1 - \frac{v^2}{c^2} = \left(\frac{3}{7} \right) \).

Step 6: Conclusion

Therefore, the relativistic speed \( v \) of the particle when its kinetic energy is 50% greater than the Newtonian value is \( v = c \sqrt{\frac{3}{7}} \).

Key Concepts

Newtonian Kinetic Energy

Newtonian kinetic energy describes the energy an object possesses due to its motion in classical mechanics. This concept is essential when dealing with objects moving at speeds much slower than the speed of light. In this regime, relativistic effects are negligible, and the kinetic energy (\[ K_n \]) is given by the familiar equation:\[ K_n = \frac{1}{2} mv^2 \]where:

• \( m \): mass of the object
• \( v \): velocity of the object

This simple equation is derived from integrating the work needed to accelerate an object from rest to a speed \( v \) and assumes that mass remains constant irrespective of the velocity at which it travels. Newtonian kinetic energy falls short when speeds approach the speed of light, requiring us to consider relativistic effects for accurate calculations.

Lorentz Factor

The Lorentz factor, denoted as \( \gamma \), is crucial in understanding relativistic physics, especially when velocities are comparable to the speed of light. As an object's velocity increases and approaches the speed of light, the Lorentz factor quantifies the degree to which Newtonian predictions need correction. Mathematically, it is expressed as:\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]Where:

• \( v \): speed of the object
• \( c \): speed of light in a vacuum (~\( 3 \times 10^8 \) meters per second)

The Lorentz factor becomes significantly larger than 1 as the speed \( v \) approaches \( c \). This results from the relativistic effects that increase an object's observed mass and impact its energy and time experienced. In the context of kinetic energy, the Lorentz factor modifies the expression, allowing us to calculate the relativistic kinetic energy for fast-moving particles.

Speed of Light

The speed of light, represented by \( c \), is a fundamental constant in physics, valued at approximately \( 3 \times 10^8 \) meters per second. It embodies the maximum speed at which information or matter can travel through space in our universe. An essential concept in Einstein’s theory of relativity, the speed of light underpins the relativistic effects perceived at velocities nearing \( c \).A key implication of the speed of light in relativity is the prediction of time dilation and length contraction. Objects communicated or observed at this speed establish a universal speed limit. No material object can reach or exceed the speed of light due to principal constraints of energy and mass that arise in special relativity.This speed limit influences equations like those for relativistic kinetic energy, where calculations ensure that objects remain subluminal, avoiding paradoxes of exceeding this universal constant. Understanding \( c \) provides insights into the boundaries of classical mechanics and guides us in traversing the realm of high-speed physics.