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Chapter 37: Problem 17

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of 0.600\(c .\) The pursuit ship is traveling at a speed of 0.800\(c\) relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the velocity of the cruiser relative to the pursuit ship be directed toward or away from the pursuit ship? (b) What is the speed of the cruiser relative to the pursuit ship?

Short Answer

Expert verified
(a) The velocity is directed away. (b) The speed is approximately 0.946c.

Step by step solution

01

Understanding the Relativity of Velocities

We must use the relativistic velocity addition formula to find the velocity of the cruiser relative to the pursuit ship. The formula is given by:\[v' = \frac{v + u}{1 + \frac{v \cdot u}{c^2}}\]where \(v\) is the speed of the cruiser relative to Tatooine (0.600c), \(u\) is the speed of the pursuit ship relative to Tatooine (0.800c), and \(c\) is the speed of light.
02

Applying the Relativistic Velocity Formula

Substitute the given values into the formula:\[v' = \frac{0.800c + 0.600c}{1 + \frac{0.800c \cdot 0.600c}{c^2}} = \frac{1.400c}{1 + 0.480}\]Simplify the expression in the denominator to get:\[v' = \frac{1.400c}{1.480}\]
03

Calculating the Relative Velocity

Perform the division to find the velocity of the cruiser relative to the pursuit ship:\[v' = \frac{1.400}{1.480} \times c \approx 0.946c\]
04

Interpreting the Result

The positive sign of the result \(0.946c\) indicates that the velocity of the cruiser relative to the pursuit ship is directed away from the pursuit ship. This means the cruiser is moving away but at a slow rate compared to their individual speeds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed of Light
The speed of light is a fundamental constant in physics, represented by the symbol \(c\). It is the maximum speed at which all energy, matter, and information in the universe can travel. The speed of light is approximately 299,792,458 meters per second. In formulaic terms, \(c\) appears frequently in Einstein's theory of relativity.
One key aspect of \(c\) is its role in limiting how fast objects can travel through space. Nothing with mass can reach or exceed the speed of light in a vacuum. This limitation underpins many modern physics theories.
In the context of relativistic velocity, \(c\) serves as a reference point. For example, speeds are often expressed as a fraction of the speed of light, such as 0.800\(c\) or 0.600\(c\). These values help to convey how close these velocities are to the ultimate speed limit of \(c\). Understanding \(c\) is crucial for grasping the implications of relativistic motion, especially when dealing with high-speed scenarios like spacecraft intercept missions.
Relative Velocity
Relative velocity is a measure of how fast one object is moving compared to another. It becomes particularly important in physics when objects are moving at significant fractions of the speed of light. At these speeds, simple addition of velocities doesn't work due to the effects of Einstein鈥檚 special theory of relativity.
The concept of relative velocity requires a formula that accounts for time and space being interconnected: the relativistic velocity addition formula. This formula is essential when calculating velocities from different reference frames moving at relativistic speeds. With the relativistic velocity addition, two velocities \(v\) and \(u\) combine as follows:
  • \( v' = \frac{v + u}{1 + \frac{v \cdot u}{c^2}} \)
This formula ensures the resulting velocity \(v'\) never exceeds the speed of light. For example, if a spacecraft is catching up to another vessel, their relative speed will be directly calculated using this formula. Understanding this allows us to predict their behavior as seen by an observer.
Velocity of Spacecrafts
The velocity of spacecrafts refers to their speed and direction as they traverse through space. In our given problem, this is particularly interesting because both the pursuit spacecraft and the cruiser are moving at speeds that are significant fractions of the speed of light.
To determine how these velocities relate to each other, we use the concept of relativistic velocity addition. As calculated, the pursuit ship has a velocity of 0.800\(c\), while the cruiser travels at 0.600\(c\). These speeds are considered relative to a stationary point, in this case, Tatooine.
The solution applied the relativistic velocity equation to find the cruiser鈥檚 speed relative to the pursuit spacecraft. The result was 0.946\(c\), indicating that while each spacecraft has its individual speed measured relative to a fixed point, their interaction, or relative velocity, shows one is still moving away, albeit slower relative to the other. Understanding these velocities and their calculations helps in predicting outcomes in high-speed pursuits and maneuvers in space explorations.

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Most popular questions from this chapter

When a particle meets its antiparticle, they annihilate each other and their mass is converted to light energy. The United States uses approximately \(1.0 \times 10^{20} \mathrm{J}\) of energy per year. (a) If all this energy came from a futuristic antimatter reactor, how much mass of matter and antimatter fuel would be consumed yearly? (b) If this fuel had the density of iron \(\left(7.86 \mathrm{g} / \mathrm{cm}^{3}\right)\) and were stacked in bricks to form a cubical pile, how high would it be? (Before you get your hopes up, antimatter reactors are a long way in the future-if they ever will be feasible.

The positive muon \(\left(\mu^{+}\right),\) an unstable particle, lives on average \(2.20 \times 10^{-6} \mathrm{s}\) (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of \(0.900 c,\) what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

Space pilot Mavis zips past Stanley at a constant speed relative to him of 0.800\(c .\) Mavis and Stanley start timers at zero when the front of Mavis's ship is directly above Stanley. When Mavis reads 5.00 s on her timer, she turns on a bright light under the front of her spaceship. (a) Use the Lorentz coordinate transformation derived in Example 37.6 to calculate \(x\) and \(t\) as measured by Stanley for the event of turning on the light. (b) Use the time dilation formula, Eq. \((37.6),\) to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of \(t\) you calculated in part (a). (c) Multiply the time interval by Mavis's speed, both as measured by Stanley, to calculate the distance she has traveled as measured by him when the light turns on. Compare to the value of \(x\) you calculated in part (a).

A proton (rest mass \(1.67 \times 10^{-27} \mathrm{kg} )\) has total energy that is 4.00 times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

Everyday Time Dilation. Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 250 \(\mathrm{m} / \mathrm{s}\) and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 4.00 \(\mathrm{h}\) . By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time? (Hint: since \(u \ll c,\) you can simplify \(\sqrt{1-u^{2} / c^{2}}\) by a binomial expansion.)
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