Question

Two particles in a high-energy accelerator experiment are approaching each other head-on, each with a speed of 0.9380c as measured in the laboratory. What is the magnitude of the velocity of one particle relative to the other?

Short answer

The magnitude of the velocity of one particle relative to the other is approximately 0.9983c.

Step-by-step solution

Identify the Problem

We need to find the relative velocity of two particles moving towards each other with given speeds using relativistic velocity addition.

Assign Known Values and Formula

The particles are moving towards each other with a speed of 0.9380c each. The relativistic velocity addition formula is needed, given by: \( v_{rel} = \frac{u + v}{1 + \frac{uv}{c^2}} \), where \( u \) and \( v \) are the speeds of the two particles as measured from the laboratory.

Set up the Equation

In our case, the speed \( u = 0.9380c \) and \( v = 0.9380c \). Substitute these into the relativistic addition formula: \[ v_{rel} = \frac{0.9380c + 0.9380c}{1 + \frac{(0.9380c)(0.9380c)}{c^2}} \].

Simplify the Numerator

Calculate the numerator of the expression: \( 0.9380c + 0.9380c = 1.8760c \).

Calculate the Denominator

Calculate the denominator: \( 1 + \frac{(0.9380)^2c^2}{c^2} = 1 + 0.879044 = 1.879044 \).

Calculate the Relative Velocity

Substitute the simplified terms back into the equation: \[ v_{rel} = \frac{1.8760c}{1.879044} \].

Final Computation

Divide the numerator by the denominator to get the final result: \[ v_{rel} \approx 0.9983c \].

Key concepts

Relative Velocity

Understanding relative velocity is crucial in high-energy physics experiments. When two objects move towards each other, the speed at which they approach one another differs from the speed measured in a stationary frame like a lab. In high-speed scenarios, such as those near the speed of light ( \( c \)), calculating this relative speed requires special consideration.

In our problem, both particles move at a speed of 0.9380c as measured from the lab. To find the relative velocity, we use the relativistic velocity addition formula:

• The formula: \( v_{rel} = \frac{u + v}{1 + \frac{uv}{c^2}} \)
• This accounts for the effects of relativity, ensuring the relative velocity never exceeds the speed of light.

This approach differs from classical physics, where you'd simply add the speeds, potentially exceeding \( c \). The application of this formula signifies a shift to considering the relativistic effects in velocity calculations.

High-Energy Physics

High-energy physics investigates particles moving at speeds close to the speed of light. In such settings, typical classical physics rules don't apply as they do at slower speeds. Instead, this domain uses relativistic physics to understand and predict particle behavior.

When dealing with particles in accelerators, like those in the original exercise, scientists must consider:

• The increased mass of particles as they approach light speed.
• The relativistic effects on time and space, requiring precise calculations.

These considerations fundamentally alter how experiments are designed and interpreted. Particles don't just move faster; they follow completely different sets of principles, affecting their interactions and how they can be harnessed in experiments.

Relativity

The theory of relativity, proposed by Albert Einstein, fundamentally alters how we understand motion and reference frames. It introduces the concept that laws of physics remain constant, but measurements of time, space, and velocity can change based on an observer's frame of reference.

Within relativity, especially the special theory of relativity relevant to our problem, a few key elements are:

• Time dilation: Time moves slower for objects moving at high speeds relative to an observer.
• Length contraction: Objects contract in the direction of motion as their speeds approach light speed.
• Relativistic momentum and energy, adjusting for massive speeds.

These elements assure that all physical observations align with the principle that nothing can move faster than light, providing essential corrections to our classical assumptions about velocity and motion.