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Ch. 36 - The Special Theory of Relativity
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 36 - The Special Theory of RelativityProblem 88
Chapter 35, Problem 88

A spaceship and its occupants have a total mass of 160,000 kg. The occupants would like to travel to a star that is 32 light-years away at a speed of 0.70c. To accelerate, the engine of the spaceship changes mass directly to energy.
(a) Estimate how much mass will be converted to energy to accelerate the spaceship to this speed.
(b) Assuming the acceleration is rapid, so the speed for the entire trip can be taken to be 0.70c, determine how long the trip will take according to the astronauts on board.

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Step 1: To solve part (a), use the relativistic kinetic energy formula to calculate the energy required to accelerate the spaceship to 0.70c. The formula for relativistic kinetic energy is: \( KE = (\gamma - 1)mc^2 \), where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \) is the Lorentz factor, \( m \) is the mass of the spaceship, \( v \) is the velocity (0.70c), and \( c \) is the speed of light.
Step 2: Substitute the given values into the Lorentz factor formula \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \). Calculate \( \gamma \) for \( v = 0.70c \). Then, substitute \( \gamma \) and the mass \( m = 160,000 \ \text{kg} \) into the kinetic energy formula to find the total energy required.
Step 3: To find the mass converted to energy, use Einstein's mass-energy equivalence formula \( E = \Delta m c^2 \), where \( \Delta m \) is the mass converted to energy. Rearrange the formula to solve for \( \Delta m \): \( \Delta m = \frac{E}{c^2} \). Substitute the energy calculated in Step 2 into this formula to estimate the mass converted.
Step 4: For part (b), calculate the time for the trip according to the astronauts on board. The proper time \( \Delta t' \) experienced by the astronauts is related to the time in the rest frame \( \Delta t \) by the time dilation formula: \( \Delta t' = \Delta t / \gamma \). First, calculate the time in the rest frame using \( \Delta t = \frac{d}{v} \), where \( d = 32 \ \text{light-years} \) and \( v = 0.70c \).
Step 5: Substitute the rest frame time \( \Delta t \) and the Lorentz factor \( \gamma \) (calculated in Step 2) into the time dilation formula \( \Delta t' = \Delta t / \gamma \) to find the time experienced by the astronauts. This will give the duration of the trip as perceived by the occupants of the spaceship.

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Key Concepts

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

Mass-Energy Equivalence

Mass-energy equivalence, expressed by Einstein's equation E=mc², states that mass can be converted into energy and vice versa. In the context of the spaceship, this principle allows us to calculate how much mass must be converted to achieve the desired speed. The energy required for acceleration can be derived from the kinetic energy formula, which incorporates relativistic effects at high speeds.
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Relativistic Kinematics

Relativistic kinematics deals with the motion of objects moving at speeds close to the speed of light (denoted as 'c'). At these speeds, time dilation and length contraction occur, affecting how time and distance are perceived by observers in different frames of reference. For the astronauts, the time taken for the journey will differ from that measured by an observer on Earth due to these relativistic effects.
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Time Dilation

Time dilation is a phenomenon predicted by the theory of relativity, where time passes at different rates for observers in different inertial frames. For the astronauts traveling at 0.70c, their onboard clocks will tick slower compared to those on Earth. This means that while the journey may take a significant amount of time from an Earth perspective, the astronauts will experience a shorter duration due to their high velocity.
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Related Practice
Textbook Question

Astronomers measure the distance to a particular star to be 6.0 light-years (1ly = distance light travels in 1 year). A spaceship travels from Earth to the vicinity of this star at steady speed, arriving in 3.25 years as measured by clocks on the spaceship. (a) How long does the trip take as measured by clocks in Earth’s reference frame? (b) What distance does the spaceship travel as measured in its own reference frame?

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Textbook Question

For a 1.0-kg mass, make a plot of the kinetic energy as a function of speed for speeds from 0 to 0.9c, using both the classical formula ( K = 1/2 mv²) and the correct relativistic formula ( K = ( γ -1)mc²).

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Textbook Question

Using Example 36–2 as a guide, show that for objects that move slowly in comparison to c, the length contraction formula is roughly ℓ ≈ ℓ₀ (1 - 1/2 v²/c²) . Use this approximation to find the “length shortening” ∆ℓ = ℓ₀ - ℓ of the train in Example 36–6 if the train travels at 100 km/h (rather than 0.92c).

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Textbook Question

A quasar emits familiar hydrogen lines whose wavelengths are 8.5% longer than what we measure in the laboratory.

(a) Using the Doppler formula for light, estimate the speed of this quasar.

(b) What result would you obtain if you used the “classical” Doppler shift discussed in Chapter 16?

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Textbook Question

A pi meson of mass mπ decays at rest into a muon (mass mμ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is Kμ = (mπ - mμ)² c² / (2mπ).

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Textbook Question

Two protons, each having a speed of 0.945c in the laboratory, are moving toward each other. Determine (a) the momentum of each proton in the laboratory, (b) the total momentum of the two protons in the laboratory, and (c) the momentum of one proton as seen by the other proton.

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