Question

It is concluded from measurements of the red shift of the emitted light that quasar \(Q_{1}\) is moving away from us at a speed of \(0.788 c\). Quasar \(Q_{2}\), which lies in the same direction in space but is closer to us, is moving away from us at speed \(0.413 c\). What velocity for \(Q_{2}\) would be measured by an observer on \(Q_{1}\) ?

Short answer

To find out the relative speed of the quasar Q2 with respect to Q1, apply the Einstein's velocity addition formula. Calculate the relative velocity using these values, and that'll give the answer needed.

Step-by-step solution

Understand the Concept of Relative Velocity

The velocity of an object as observed from a different moving object (also called relative velocity) can be calculated. In classical physics, this is simple subtraction. However, when objects are moving with velocity close to the speed of light, this classical formula fails and we should use the formula from the theory of relativity.

Apply the Einstein's Velocity Addition Formula

The Einstein's velocity addition formula is written as \(v_{r} = \frac{v_{1} + v_{2}}{1 + \frac{v_{1}v_{2}}{c^{2}}}\), where \(v_{r}\) is the relative velocity of the two objects, \(v_{1}\) is the velocity of the first object, \(v_{2}\) is the velocity of the second object. Here, \(v_{1} = 0.788 c\) (velocity of \(Q_{1}\)) and \(v_{2} = -0.413 c\) (opposite direction of velocity of \(Q_{2}\)).

Calculate Relative Velocity

Now that we know the velocities of \(Q_{1}\) and \(Q_{2}\) and their directions, we can get the relative velocity \(v_{r} = \frac{0.788c - 0.413c}{1 - \frac{0.788c * -0.413c}{c^{2}}}\). Solving this equation will give us the relative velocity of \(Q_{2}\) with respect to \(Q_{1}\).

Key concepts

Relative Velocity

In the realm of physics, the concept of relative velocity is crucial for understanding how different observers perceive the motion of objects. Imagine you are on a train moving at a constant speed, and you see another train passing by. The speed of this other train, as you observe it, is known as its relative velocity. In simple terms, relative velocity is the velocity of an object from the perspective of a moving observer.

• In classical physics, relative velocity is a straightforward calculation, often computed by simple subtraction of velocities.
• However, when dealing with speeds close to the speed of light, as in our case with quasars, classical methods are inadequate.
• Einstein's Theory of Relativity offers a more precise approach to handle such high-speed observations.

Understanding the relative velocity of objects moving at relativistic speeds (near the speed of light) requires using the relativity principles to get accurate results.

Velocity Addition Formula

When handling speeds approaching the speed of light, the velocity addition formula from Einstein's theory is indispensable. Unlike the linear velocity addition used in classical physics, Einstein's formula provides a way to calculate relative velocities for high-speed celestial objects like quasars.

The formula is:

• \[ v_{r} = \frac{v_{1} + v_{2}}{1 + \frac{v_{1}v_{2}}{c^{2}}} \]
• Here, \(v_{r}\) is the relative velocity, \(v_{1}\) and \(v_{2}\) are the velocities of two objects.
• \(c\) represents the speed of light.

This formula accounts for the fact that no object can exceed the speed of light, aligning with the postulates of relativity. When applied to quasars, it ensures accurate relative velocity computation, using their significant speeds.

Red Shift

Red shift is a fascinating phenomenon that provides key insights into the universe's vastness and movement. When an object in space moves away from us, the light it emits shifts towards the longer wavelength (red) end of the spectrum. This change is known as red shift, and it is crucial for measuring cosmic velocities.

• Reliable for detecting high-speed celestial motion, such as quasars traveling at relativistic speeds.
• Direct evidence of an object's speed along the line of sight, given as a shift in the observed electromagnetic spectrum.
• Also serves as a tool to gauge the expansion of the universe, where further red shifts imply faster recession speeds.

In the exercise, red shift allows us to determine quasars' velocities relative to Earth, shining light on their cosmic distances and velocities.

Speed of Light

The speed of light is fundamental in physics, serving as the cosmic speed limit of the universe. At approximately 299,792,458 meters per second, or about 300,000 kilometers per second, it forms the benchmark for relativistic calculations. This constant, denoted as \(c\), is pivotal in Einstein's relativity equations.

• Serves as the upper limit for how fast information or matter can travel.
• Plays a central role in the velocity addition formula, ensuring that combined velocities do not exceed \(c\).
• Influences phenomena such as time dilation and length contraction in relativistic scenarios.

Understanding the speed of light's constraint is crucial in appreciating Einstein’s theory and resolving high-velocity astrophysical problems like the quasar exercise. It underscores the relativistic effects causing deviations from classical physics predictions.