Question

If spaceship \(A\) is traveling at \(70 \%\) of the speed of light relative to an observer at rest, and spaceship \(\mathrm{B}\) is traveling at \(90 \%\) of the speed of light relative to an observer at rest, which of the following have the greatest velocity as measured by an observer in spaceship B? a) a cannonball shot from \(A\) to \(B\) at \(50 \%\) of the speed of light as measured in A's reference frame b) a ball thrown from \(B\) to \(A\) at \(50 \%\) of the speed of light as measured in B's reference frame c) a particle beam shot from a stationary observer to \(\mathrm{B}\) at \(70 \%\) of the speed of light as measured in the stationary reference frame d) a beam of light shot from A to B, traveling at the speed of light in A's reference frame e) All of the above have the same velocity as measured in B's reference frame.

Short answer

Answer: The beam of light shot from spaceship A to spaceship B has the greatest velocity in spaceship B's frame, with a velocity of 0.9836c.

Step-by-step solution

Understand the relativistic addition of velocities formula

The relativistic addition of velocities formula is given by: \(v_{AB} = \frac{v_{A} + v_{B}}{1 + \frac{v_{A}v_{B}}{c^2}}\) where \(v_{AB}\) is the velocity of object A as measured by an observer in frame B, \(v_A\) is the velocity of object A in its own frame, \(v_B\) is the velocity of frame B relative to the stationary observer, and \(c\) is the speed of light.

Convert the given velocities as percentages of the speed of light

We are given that spaceship A is moving at \(70\%\) of the speed of light (\(0.7c\)) and spaceship B is moving at \(90\%\) of the speed of light (\(0.9c\)). We also have the velocities of objects with respect to A's and B's reference frames, which are also given in percentages of the speed of light. Convert these percentages into decimals and multiply by the speed of light \(c\) to get the velocities in the form of \(vc\).

Calculate the velocities in B's frame using the relativistic addition formula

Let's calculate the velocities of the objects in B's frame (spaceship B) using the relativistic addition formula for each case mentioned earlier. a) \(v_{AB} = \frac{0.7c + 0.5c}{1 + \frac{(0.7c)(0.5c)}{c^2}}\) b) \(v_{BA} = \frac{-0.9c + 0.5c}{1 + \frac{(-0.9c)(0.5c)}{c^2}}\) c) \(v_{BB'} = \frac{0.7c + (-0.9c)}{1 + \frac{(0.7c)(-0.9c)}{c^2}}\) d) \(v_{AB'} = \frac{0.7c + c}{1 + \frac{(0.7c)(c)}{c^2}}\)

Compare the calculated velocities

Now, we will compare the calculated velocities of objects in B's reference frame to find out which one has the greatest velocity. a) \(v_{AB} = 0.9231c\) b) \(v_{BA} = -0.6471c\) c) \(v_{BB'} = -0.2771c\) d) \(v_{AB'} = 0.9836c\)

Choose the object with the greatest velocity and present the answer

From the calculated velocities, we can see that the object with the greatest velocity in spaceship B's frame is the beam of light shot from A to B (option d), which has a velocity of \(0.9836c\). Therefore, the correct answer is option d.

Key concepts

Special Relativity

Special Relativity is a foundational theory in physics formulated by Albert Einstein. It introduced revolutionary concepts about the nature of space and time and is centered around two postulates: the laws of physics are invariant (identical) in all inertial frames of reference, and the speed of light in a vacuum is the same for all observers, regardless of their relative motion.

The theory has several implications, one of which is the relativistic addition of velocities. This arises when one tries to determine how fast an object is moving relative to another object, when both are moving at a significant fraction of the speed of light. According to special relativity, we cannot simply add their speeds together using classical physics, due to time dilation and length contraction. These effects become more pronounced as objects move faster, approaching the speed of light.

The usual intuition from daily experiences doesn't apply at relativistic speeds, which is why we use the relativistic formula for adding velocities to find the speed of one object as observed from another moving object.

Speed of Light

The speed of light in a vacuum, commonly denoted as 'c', is approximately 299,792,458 meters per second, and it's a fundamental constant in physics. According to special relativity, it's the maximum speed at which all energy, matter, and information in the universe can travel.

It's important to understand that the speed of light is not just a high velocity; it's a critical component of the structure of space and time themselves. The constancy of the speed of light leads to effects like time dilation, where time can actually pass at different rates depending on the relative speeds of observers.

It's also crucial in the relativistic addition of velocities, as seen in our exercise. The reason we don't experience these relativistic effects in our everyday life is that the speeds we encounter are very low compared to the speed of light. Nonetheless, when dealing with high-speed objects, such as particles in accelerators or cosmic velocities, the rules of special relativity firmly come into play.

Reference Frame

A reference frame, or frame of reference, is a system for specifying the position and direction of objects in space at a specific time. In physics, it's a set of coordinates that can be used to determine positions and velocities of objects. Every observation of an object's speed and position is made relative to a particular frame of reference.

An inertial reference frame is one in which the observer is not accelerating (i.e., moving at a constant velocity). Special relativity focuses on such frames because, inside them, the laws of physics have simpler and more universal forms. A non-inertial frame of reference is one where the observer is experiencing acceleration.

In the context of the exercise, when calculating the velocities of the ball, cannonball, and particle beam, one must be careful to consider the reference frames of both the observer and the moving objects. Spaceship A and B, as well as a stationary observer, each constitute different reference frames. Misunderstandings about the movement of objects often arise from neglecting to consider the appropriate reference frame, so it's essential to clearly identify it in any analysis of motion, especially in relativistic contexts.