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Chapter 27: Problem 38

A space probe is sent to the vicinity of the star Capella, which is 42.2 light-years from the earth. The probe travels with a speed of \(0.9910 c\) relative to the earth. An astronaut recruit on board is 19 years old when the probe leaves the earth. What is her biological age when the probe reaches Capella, as measured by (a) the astronaut and (b) someone on earth?

Short Answer

Expert verified
a) 25 years b) 61.6 years

Step by step solution

01

Identify the Known Variables

The distance to Capella is 42.2 light-years, and the speed of the probe is given as \(0.9910c\). The initial age of the astronaut is 19 years. We need to find her age when the probe reaches Capella, both from the perspective of the astronaut and from someone on Earth.
02

Calculate the Time from Earth's Perspective

Time taken as perceived from Earth \(t_{earth}\) is calculated using the formula for speed: \( \text{distance} = \text{speed} \times \text{time} \). So, \( t_{earth} = \frac{42.2 \text{ light-years}}{0.9910c} \). This simplifies to:\[ t_{earth} = \frac{42.2}{0.9910} \approx 42.6 \text{ years} \]
03

Calculate Time Dilation for the Astronaut

Due to time dilation effects described by special relativity, the time experienced by the astronaut \(t_{astro}\) is shorter. The time dilation formula is:\[ t_{astro} = \frac{t_{earth}}{\gamma} \] where the Lorentz factor \(\gamma\) is given by:\[ \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \] Plugging in \(v = 0.9910c\):\[ \gamma = \frac{1}{\sqrt{1 - (0.9910)^2}} \approx 7.09 \]Therefore:\[ t_{astro} = \frac{42.6}{7.09} \approx 6.01 \text{ years} \]
04

Calculate the Astronaut's Age from Her Perspective

Initially, the astronaut is 19 years old. Adding the time experienced by the astronaut \(t_{astro} = 6.01\):\[ \text{Astronaut's age} = 19 + 6.01 \approx 25.01 \text{ years} \]
05

Calculate the Astronaut's Age from Earth's Perspective

As per Earth鈥檚 time \(t_{earth} = 42.6\) years, the astronaut ages those many years during the probe's trip. Thus:\[ \text{Astronaut's age seen from Earth} = 19 + 42.6 = 61.6 \text{ years} \]

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Key Concepts

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

Time Dilation
Time dilation is a fascinating phenomenon predicted by Albert Einstein's theory of special relativity. It occurs when an object moves at a significant fraction of the speed of light, causing time to pass differently for it compared to a stationary observer. In this exercise, the astronaut onboard the fast-moving probe experiences less passage of time compared to someone on Earth, making time appear to "dilate." This explains why the astronaut ages only about 6 years while someone on Earth perceives her trip taking over 42 years.
The formula for time dilation is given by:
  • \[ t_{astro} = \frac{t_{earth}}{\gamma} \]
  • Where \( t_{astro} \) is the time experienced by the astronaut, and \( t_{earth} \) is the time observed on Earth.
This concept underscores how physical phenomena can differ significantly depending on the observer's frame of reference, highlighting the relative nature of time.
Lorentz Factor
The Lorentz factor, denoted by \( \gamma \), is crucial in understanding time dilation and other relativistic effects. It quantifies the amount by which time, length, and relativistic mass change for an object moving at relativistic speeds. Calculated using the formula:
  • \[ \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]
  • Where \( v \) is the speed of the object, and \( c \) is the speed of light.
For the probe travelling at a speed of \( 0.9910c \), the Lorentz factor is approximately 7.09. This means that time, from the astronaut's perspective, runs about seven times slower compared to the perspective of someone on Earth.
Understanding the Lorentz factor is essential to make sense of how time dilation and other relativistic effects occur.
Light-year Distance
A light-year is a unit of distance that represents how far light travels in one year. Since light travels incredibly fast鈥攁bout 299,792 kilometers per second鈥攁 light-year is quite a vast distance. In this exercise, the distance to the star Capella is given as 42.2 light-years.
This helps to illustrate the immense scales involved in space travel and the challenge in traversing these distances at anything but relativistic speeds.
  • 1 light-year = approximately 9.46 trillion kilometers.
  • Large distances in astronomy are conveniently measured in light-years because they align with the universal constant, the speed of light.
Understanding light-year as a distance measure can help students visualize and comprehend just how far stars really are from us.
Relativistic Speeds
Relativistic speeds are significant fractions of the speed of light, where effects predicted by Einstein's theory of relativity become prominent. At these speeds, traditional Newtonian physics fails to accurately describe what happens.
In the exercise, the probe travels at \( 0.9910c \), 99.10% the speed of light. Such high velocities bring about phenomena like time dilation and length contraction.
  • Effects associated with relativistic speeds include time dilation, length contraction, and increased relativistic mass.
  • The closer an object's speed is to the speed of light, the more pronounced these relativistic effects become.
Recognizing these unique characteristics at high velocities is essential for understanding how the universe operates at its fundamental levels.

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Most popular questions from this chapter

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is \(0.650 c,\) and the speed of each particle relative to the other is \(0.950 c .\) What is the speed of the second particle, as measured in the laboratory?

An enemy spaceship is moving toward your starfighter with a speed of \(0.400 c,\) as measured in your reference frame. The enemy ship fires a missile toward you at a speed of \(0.700 c\) relative to the enemy ship. (See Figure \(27.24 .)\) (a) What is the speed of the missile relative to you? Express your answer in terms of the speed of light. (b) If you measure the enemy ship to be \(8.00 \times 10^{6} \mathrm{~km}\) away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you?

Sketch a graph of (a) the nonrelativistic Newtonian momentum as a function of speed \(v\) and (b) the relativistic momentum as a function of \(v .\) In both cases, start from \(v=0\) and include the region where \(v \rightarrow c\). Does either of these graphs extend beyond \(v=c ?\)

When a particle meets its antiparticle (more about this in Chapter 30 ), 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 would be consumed yearly? (b) If this antimatter fuel had the density of Fe \(\left(7.86 \mathrm{~g} / \mathrm{cm}^{3}\right)\) 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.)

Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging. (a) If each swing takes \(1.50 \mathrm{~s}\) as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment? (b) If each swing takes \(1.50 \mathrm{~s}\) as measured by a person at mission control on earth, how long will it take as measured by the astronaut in the spaceship?
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