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Chapter 35: Problem 48

A rocket ship approaching Earth at \(0.90 c\) fires a missile toward Earth with a speed of \(0.50 c\), relative to the rocket ship. As viewed from Earth, how fast is the missile approaching Earth?

Short Answer

Expert verified
Answer: The missile is approaching Earth at a velocity of approximately 0.9655c, as seen by an observer on Earth.

Step by step solution

01

Write down the required formula

To calculate the speed of the missile as observed from Earth, we need to use the relativistic addition of velocities formula given by: \(v_r = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}\) where \(v_r\) = resultant velocity (which is the observed velocity of the missile relative to Earth), \(v_1\) = velocity of the rocket ship relative to the Earth, \(v_2\) = velocity of the missile relative to the rocket ship, \(c\) = speed of light.
02

Identify the given values

We are given, \(v_1\) = 0.90c (velocity of the rocket ship approaching Earth), \(v_2\) = 0.50c (velocity of the missile relative to the rocket ship), and \(c\) = speed of light. We need to find \(v_r\) (velocity of the missile approaching Earth, relative to Earth).
03

Substitute the given values into the formula

Let's substitute the given values into the formula: \(v_r = \frac{(0.90c) + (0.50c)}{1 + \frac{(0.90c)(0.50c)}{c^2}}\)
04

Cancel out and simplify the formula

Cancel out the common term \(c\) in the numerator and denominator: \(v_r = \frac{0.90 + 0.50}{1 + \frac{0.90(0.50)}{1}}\) Now, simplify the formula to get: \(v_r = \frac{1.40}{1 + 0.45}\)
05

Solve for the resultant velocity \(v_r\)

Calculate \(v_r\) by solving the expression: \(v_r = \frac{1.40}{1.45}\) There's no need to convert back to units of \(c\) since the answer is required in units of \(c\). \(v_r \approx 0.9655c\)
06

Conclusion

The missile is approaching Earth at a velocity of approximately \(0.9655c\) as observed from Earth.

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Key Concepts

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

Special Relativity
When we study motion at speeds comparable to the speed of light, we have to use the principles of special relativity, a theory propounded by Albert Einstein in 1905. Unlike classical mechanics, which works under the assumption that time and space are absolute, special relativity introduces the concept that time and space are relative to the observer. This means that time can dilate—run slower—and lengths can contract depending on the speed of the observer relative to the speed of the object being observed.

Moreover, special relativity introduces the idea that no object with mass can travel at the speed of light in vacuum, and as objects approach this speed, their mass effectively becomes infinite. Hence, we cannot use the standard addition of velocities as done in classical mechanics. Instead, the relativistic addition of velocities formula is used to accurately calculate the velocities of objects moving at speeds close to the speed of light.
Resultant Velocity
The resultant velocity in relativistic terms is quite different from classical predictions. It's the effective velocity of an object as measured by an observer, considering the relative motion of both. In classical mechanics, you would simply add the velocities of two objects to determine the resultant velocity. However, this doesn't hold up when dealing with speeds close to the speed of light.

In our scenario, the rocket ship and the missile each have a velocity relative to the Earth and to each other, respectively. To figure out how fast the missile is moving relative to Earth, we must employ the relativistic formula for the addition of velocities: \[v_r = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}\] This formula accounts for the fact that as objects move faster, the way their velocities combine also changes due to the nature of spacetime as described by special relativity.
Speed of Light
The speed of light, commonly denoted as \(c\), is a universal constant and is approximately 299,792,458 meters per second (or about 186,282 miles per second). This is not just the speed at which light travels; it's also the maximum speed at which all matter and information in the universe can travel, according to the theory of special relativity.

Unalterable Speed Limit

As an unalterable speed limit of the universe, the speed of light plays a crucial role in relativistic physics. It is essential in calculating the relativistic effects on time, space, and mass. For instance, our intuition might suggest that if a rocket ship is moving at \(0.90c\) towards the Earth and fires a missile at \(0.50c\) in the same direction, the speeds should add up classically to \(1.40c\). However, due to the relativistic nature of space and time, this isn't the case, and the actual resultant velocity is obtained by using the relativistic velocity addition formula, showing us that we cannot exceed the speed of light, even when adding velocities that individually are below \(c\).

Understanding these concepts allows students to navigate the counterintuitive results that often arise when studying high-velocity scenarios in the universe.

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Most popular questions from this chapter

Use light cones and world lines to help solve the following problem. Eddie and Martin are throwing water balloons very fast at a target. At \(t=-13 \mu s,\) the target is at \(x=0,\) Eddie is at \(x=-2 \mathrm{~km},\) and Martin is at \(x=5 \mathrm{~km} ;\) all three remain in these positions for all time. The target is hit at \(t=0 .\) Who made the successful shot? Prove this using the light cone for the target. When the target is hit, it sends out a radio signal. When does Martin know the target has been hit? When does Eddie know the target has been hit? Use the world lines to show this. Before starting to draw the diagrams, consider this: If your \(x\) position is measured in kilometers and you are plotting \(t\) versus \(x / c,\) what unit must \(t\) be in?

Consider motion in one spatial dimension. For any velocity \(v_{n}\) define the parameter \(\theta\) via the relation \(v=c \tanh \theta,\) where \(c\) is the speed of light in vacuum. This quantity is called the velocity parameter or the rapidity corresponding to velocity \(v\). a) Prove that for two velocities, which add via a Lorentz transformation, the corresponding velocity parameters simply add algebraically, that is, like Galilean velocities. b) Consider two reference frames in motion at speed \(v\) in the \(x\) -direction relative to one another, with axes parallel and origins coinciding when clocks at the origin in both frames read zero. Write the Lorentz transformation between the two coordinate systems entirely in terms of the velocity parameter corresponding to \(v\) and the coordinates.

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.

As the velocity of an object increases, so does its energy. What does this imply? a) At very low velocities, the object's energy is equal to its mass times \(c^{2}\). b) In order for an object with mass \(m\) to reach the speed of light, infinite energy is required. c) Only objects with \(m=0\) can travel at the speed of light. d) all of the above e) none of the above

A meteor made of pure kryptonite (yes, we know that there really isn't such a thing as kryptonite ... ) is moving toward Earth. If the meteor eventually hits Earth, the impact will cause severe damage, threatening life as we know it. If a laser hits the meteor with light of wavelength \(560 \mathrm{nm}\), the meteor will blow up. The only laser on Earth powerful enough to hit the meteor produces light with a 532 -nm wavelength. Scientists decide to launch the laser in a spacecraft and use special relativity to get the right wavelength. The meteor is moving very slowly, so there is no correction for relative velocities. At what speed does the spaceship need to move so that the laser light will have the right wavelength, and should it travel toward or away from the meteor?
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