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Chapter 2: Problem 40

An experimenter determines that a particle created at one end of the laboratory apparatus moved at \(0.94 c\) and survived for 0.032 \mus, decaying just as it reached the other end. (a) According to the experimenter. how far did the particle move? (b) In its own frame of reference, how long did the particle survive? (c) According to the particle, what was the length of the laboratory apparatus?

Short Answer

Expert verified
The particle moved approximately \(8957.6\) m according to the experimenter. In its own frame of reference, the particle says it endured for about \(0.1\) \mus. Finally, according to the particle, the laboratory apparatus is around \(2997.2\) m in length.

Step by step solution

01

Calculation of Distance Travelled

To find out how far the particle had traveled, multiply the speed of the particle (\(0.94 c\)) by the time it had traveled (0.032 \mus). Since \(c\) stands for the speed of light (\(3 \times 10^{8}\) m/s), convert \mus to seconds before the multiplication.
02

Calculation of Proper Time

The proper time is the time measured in the rest frame of the event. Here, the proper time is the time it survived according to the particle. As the experimenter's time will be dilated due to the particle's speed, use the formula for time dilation which is \(T = T_{0} / sqrt(1 - v^2 / c^2)\), where \(T_{0}\) is the proper time, \(v\) is the velocity of the particle, \(c\) is the speed of light, and \(T\) is the dilated time. Rearrange the formula and solve for \(T_{0}\).
03

Calculation of Proper Length

The proper length is the length measured in the rest frame of the object. Here, the proper length is the length of the laboratory apparatus according to the particle. Since the experimenter's length is contracted due to the particle's speed, use the formula for length contraction, which is \(L = L_{0} \times sqrt(1 - v^2 / c^2)\), where \(L_{0}\) is the proper length, \(v\) is the velocity of the particle, and \(c\) is the speed of light. Again, rearrange the formula and solve for \(L_{0}\).

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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 key concept in relativistic physics. It explains why a moving clock ticks slower compared to a stationary one. Imagine you're watching a fast-moving spaceship with a clock on board. From your perspective, the clock ticks slower than a clock at rest beside you.
This happens because, according to Einstein's theory of relativity, time is not absolute. When an object moves close to the speed of light, time for that object stretches out or "dilates."
  • The formula used to calculate time dilation is \( T = \frac{T_0}{\sqrt{1 - \frac{v^2}{c^2}}} \).
  • Here, \( T \) is the time you measure, \( T_0 \) is the proper time (the time measured in the moving object's frame), \( v \) is the velocity, and \( c \) is the speed of light.
This means, for the particle moving at 0.94 times the speed of light, its lifespan appears longer to the experimenter than it does for the particle itself.
Length Contraction
Length contraction refers to how objects appear shorter when they move at high speeds, nearing the speed of light. If you see a spaceship zoom past you, it would seem shorter than when it's at rest.
This contraction only happens in the direction of motion and is a result of the relativistic principles.
  • The formula for length contraction is \( L = L_0 \times \sqrt{1 - \frac{v^2}{c^2}} \).
  • \( L \) is the contracted length observed, \( L_0 \) is the proper length (measured when the object is at rest), \( v \) is the velocity, and \( c \) is the speed of light.
In the given scenario, the lab apparatus appears shorter from the particle’s perspective because it's moving at a high speed.
Speed of Light
The speed of light, denoted as \( c \), is a universal constant in physics, valued at approximately \( 3 \times 10^8 \) meters per second. It represents the maximum speed at which all energy, matter, and information can travel.
Light travels at this speed in a vacuum, and it forms the backbone of Einstein's theory of relativity.
  • It's key to note that as an object approaches the speed of light, relativistic effects like time dilation and length contraction become significant.
  • Nothing in the universe can exceed this speed, making it a natural speed limit.
In the exercise, the particle is moving at 0.94 \( c \), highlighting just how close it is to this universal limit.
Proper Time
Proper time is the time measured in the rest frame of a moving object. It is the time as experienced by someone traveling with the object.
For example, if you're sitting in a spaceship traveling at a high speed, your watch measures the proper time because you're at rest relative to your surroundings.
  • In mathematical terms, proper time is \( T_0 \) in the time dilation formula.
  • Proper time is crucial for calculations in relativistic physics, like determining how long particles last in their own frame of reference.
In the problem, the proper time is the duration of the particle’s life in its own frame, which we need to compute using the time dilation formula.

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Most popular questions from this chapter

In the collision shown, energy is conserved. because both objects have the same speed and mass after as before the collision. Since the collision merely reverses the velocities, the final (total) momentum is opposite the initial. Thus, momentum can be conserved ooly if it is zero. (a) Using the relativistically correct expression for momentum, show that the total momentum is zero that momentum is conserved. (Masses are in arbitrary units.) (b) Using the relativistic velocity transformation, find the four velocities in a frame moving to the right at \(0.6 c\) (c) Verify that momentum is conserved in che new frame.

A \(3.000\) u ob ject moving to the right through a laboratory at \(0.8 c\) collides with a \(4.000\) u ob ject moving to the left through the laboratory at \(0.6 c\). Afterward, there are two objects. one of which is a \(6.000\) u mass at rest. (a) What are the mass and speed of the other object? (b) Determine the change in kinetic energy in this collision.

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You are floating in space when you notice a flying saucer circling you. Each time it passes in front of you, you note the reading on its clock. Do you see its clock advancing faster or slower than your wristwatch? Does the space alien see your wristwatch advancing faster or slower than his clock? Explain.
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