Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.650c, and the speed of each particle relative to the other is 0.950c. What is the speed of the second particle, as measured in the laboratory?

Verified step by step guidance

1. Start by identifying the relevant concept: This problem involves relativistic velocity addition, which is used when dealing with velocities close to the speed of light (c). The formula for relativistic velocity addition is: $u\text{'}=\frac{u+v}{1+\frac{uv}{c^2}}$, where u' is the relative velocity, u is the velocity of one object, and v is the velocity of the other object in the same reference frame.
2. Assign the given values: The speed of the first particle relative to the laboratory is $0.650c$, and the relative speed between the two particles is $0.950c$. Let the speed of the second particle relative to the laboratory be $v$. Substitute these into the relativistic velocity addition formula.
3. Rearrange the formula to solve for the unknown velocity $v$: The relative velocity between the two particles is given as $0.950c$, so substitute $u\text{'}=0.950c$ and $u=0.650c$ into the formula. Solve for $v$.
4. Simplify the equation: Substitute the known values into the denominator as well. The denominator becomes $1+\frac{0.650c0.950c}{c^2}$. Since $c$ cancels out, the denominator simplifies to $1+0.6175$. Use this to simplify the equation further.
5. Solve for $v$: After simplifying the numerator and denominator, isolate $v$ to find the speed of the second particle relative to the laboratory. This will give you the final expression for $v$ in terms of $c$.

Key Concepts

Relativity of Velocity

In special relativity, the velocity of an object as observed from different reference frames can differ significantly. The relativistic velocity addition formula is used to calculate the resultant speed of one object as observed from another moving object. This concept is crucial for understanding how speeds combine when dealing with velocities close to the speed of light.

Lorentz Factor

The Lorentz factor, denoted as gamma (γ), is a key component in special relativity that accounts for time dilation and length contraction at relativistic speeds. It is defined as γ = 1 / √(1 - v²/c²), where v is the object's speed and c is the speed of light. This factor becomes significant as speeds approach the speed of light, affecting measurements of time and distance.

Invariant Speed of Light

One of the fundamental postulates of special relativity is that the speed of light in a vacuum is constant and invariant, regardless of the observer's motion. This principle leads to counterintuitive results, such as time dilation and length contraction, and is essential for understanding how particles behave at high speeds, particularly in high-energy physics experiments.