Ch. 36 - The Special Theory of Relativity

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 36 - The Special Theory of Relativity

Problem 18

Chapter 35, Problem 18

(I) An observer on Earth sees an alien vessel approach at a speed of 0.70c. The fictional starship Enterprise comes to the rescue (Fig. 36–17), overtaking the aliens while moving directly toward Earth at a speed of 0.90c relative to Earth. What is the relative speed of one vessel as seen by the other?

Verified step by step guidance

Step 1: Identify the problem as a relativistic velocity addition problem. In special relativity, when two objects are moving relative to each other at significant fractions of the speed of light, their relative velocity cannot be calculated using classical mechanics. Instead, we use the relativistic velocity addition formula.

Step 2: Write down the relativistic velocity addition formula: \( v_{rel} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \), where \( v_1 \) and \( v_2 \) are the velocities of the two objects relative to a common reference frame, and \( c \) is the speed of light.

Step 3: Assign the given values: \( v_1 = 0.90c \) (Enterprise's speed relative to Earth) and \( v_2 = -0.70c \) (alien vessel's speed relative to Earth, negative because it is moving toward Earth). Substitute these values into the formula.

Step 4: Simplify the numerator: \( v_1 + v_2 = 0.90c - 0.70c = 0.20c \). Then calculate the denominator: \( 1 + \frac{v_1 v_2}{c^2} = 1 + \frac{(0.90c)(-0.70c)}{c^2} = 1 - 0.63 = 0.37 \).

Step 5: Combine the results to find the relative velocity: \( v_{rel} = \frac{0.20c}{0.37} \). This gives the relative speed of one vessel as seen by the other. You can simplify further if needed, but the process is complete.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relativity of Velocity

In the context of special relativity, the relative velocity between two objects moving at significant fractions of the speed of light (c) cannot be calculated using classical addition of velocities. Instead, the relativistic velocity addition formula must be used, which accounts for the effects of time dilation and length contraction as objects approach the speed of light.

Lorentz Transformation

The Lorentz transformation equations relate the space and time coordinates of events as observed in different inertial frames. These transformations are essential for understanding how measurements of time and distance change for observers moving relative to one another, particularly at high speeds close to the speed of light.

Speed of Light (c)

The speed of light in a vacuum, denoted as c, is a fundamental constant in physics, approximately equal to 299,792,458 meters per second. It serves as the ultimate speed limit in the universe, influencing the behavior of objects moving at relativistic speeds and forming the basis for Einstein's theory of relativity, which reshapes our understanding of space and time.

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Related Practice

Textbook Question

A spaceship traveling at 0.76c away from Earth fires a module with a speed of 0.85c at right angles to its own direction of travel (as seen by the spaceship). What is the speed of the module, and its direction of travel (relative to the spaceship’s direction), seen by an observer on Earth?

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Textbook Question

Two spaceships leave Earth in opposite directions, each with a speed of 0.50c with respect to Earth.

(a) What is the velocity of spaceship 1 relative to spaceship 2?

(b) What is the velocity of spaceship 2 relative to spaceship 1?

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Textbook Question

A star is 23.5 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers on the spacecraft?

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Textbook Question

(I) Repeat Problem 19 using the Lorentz transformation and a relative speed v = 1.60 x 10⁸ m/s, but choose the time t to be (a) 3.5μs and (b) 10.0 μs .

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