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Chapter 26: Problem 17

A pole vaulter at the Relativistic Olympics sprints past you to do a vault at a speed of \(0.65 c\). When he is at rest, his pole is \(7.0 \mathrm{~m}\) long. (a) What length do you perceive the pole to be as he passes you, assuming his relative velocity is parallel to the length of the pole? (b) How long does it take the pole to pass a given location on the track according to a track-based observer? (c) Repeat part (b) from the vaulter's reference frame and explain why the two answers are different.

Short Answer

Expert verified
(a) 5.32 m; (b) 2.73 x 10^-8 s; (c) 3.59 x 10^-8 s; due to relativity of simultaneity differences.

Step by step solution

01

Understanding Lorentz Contraction

When an object moves at relativistic speeds parallel to its length, its length is contracted in the frame of a stationary observer. This phenomenon is known as Lorentz contraction. The formula to calculate the contracted length is given by: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where \( L \) is the observed length, \( L_0 \) is the rest length (7.0 m in this case), \( v \) is the velocity of the object (0.65c), and \( c \) is the speed of light.
02

Calculating the Contracted Length

Substitute the given values into the Lorentz contraction formula:\[ L = 7.0 \text{ m} \times \sqrt{1 - \left(0.65\right)^2} \]Calculate \( \left(0.65\right)^2 = 0.4225 \), and then \( 1 - 0.4225 = 0.5775 \). The square root of 0.5775 is approximately 0.7596. Thus, the contracted length is:\[ L \approx 7.0 \text{ m} \times 0.7596 \approx 5.32 \text{ m} \].
03

Time to Pass a Location for Track-Based Observer

The time taken to pass a given location on the track for the stationary observer can be calculated using the formula:\[ t = \frac{L}{v} \]where \( L \) is the contracted length (5.32 m) and \( v \) is the velocity (0.65c). First, convert \( v = 0.65c \) to meters per second, \( 0.65 \times 3 \times 10^8 \text{ m/s} = 1.95 \times 10^8 \text{ m/s} \). Now calculate the time:\[ t = \frac{5.32 \text{ m}}{1.95 \times 10^8 \text{ m/s}} \approx 2.73 \times 10^{-8} \text{ s} \].
04

Time to Pass a Location for Vaulter's Frame

From the vaulter's frame, the pole does not undergo Lorentz contraction and remains at its rest length of 7.0 m. The time, therefore, is:\[ t = \frac{7.0 \text{ m}}{1.95 \times 10^8 \text{ m/s}} \approx 3.59 \times 10^{-8} \text{ s} \].
05

Explaining the Difference

The time difference arises due to the relativity of simultaneity, a key concept in special relativity. In the track observer's frame, the pole is shorter due to Lorentz contraction, resulting in a shorter time to pass. However, in the vaulter's frame, the pole remains at its full length, which is why it takes longer to pass.

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Key Concepts

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

Relativistic Speeds
When objects travel at speeds close to the speed of light, strange but fascinating effects occur. These speeds are known as "relativistic speeds." To get an idea of what this means, consider the speed of light in a vacuum, which is approximately \(3 imes 10^8\) meters per second, a mind-boggling speed that we don't encounter in everyday life.

At relativistic speeds, the regular laws of motion from Newtonian physics no longer apply the same way. Instead, we have to consider the framework of Einstein's theory of special relativity. This framework accounts for changes in the length, mass, and even the perception of time for objects as they move extremely fast.

For a pole vaulter sprinting with a speed of \(0.65c\), this means that not only is he moving incredibly fast, but also that strange phenomena such as Lorentz contraction will influence how observers perceive the length of his pole.
Special Relativity
Special relativity is an essential theory in modern physics proposed by Albert Einstein in 1905. It revolutionized our understanding of space, time, and motion at high speeds, near the speed of light.

The key principles of special relativity are:
  • The laws of physics are the same for all observers, regardless of their relative motion.
  • The speed of light is the same for all observers, no matter how fast they are moving relative to the light source.

This results in several counterintuitive effects, such as:
  • Time dilation: Moving clocks run slower when observed from a stationary reference frame.
  • Lorentz contraction: Fast-moving objects appear shortened along the direction of their movement from a stationary observer's viewpoint.
  • Relativity of simultaneity: Events that happen at the same time in one frame may not be simultaneous in another.
In the exercise, the pole vaulter's pole seemed to contract due to the effects of Lorentz contraction, a direct result of the principles of special relativity.
Relativity of Simultaneity
A particularly interesting and somewhat mind-bending aspect of special relativity is the relativity of simultaneity. This concept tells us that simultaneity is not an absolute concept but rather relative depending on the observer's state of motion.

Imagine two events happening simultaneously from your point of view. However, for an observer moving at a high speed relative to your frame, these events might not occur at the same time. This discrepancy is a key highlighting feature of special relativity and stems from the fact that time is intertwined with space in a complex way.

In the context of the pole vaulter scenario, the observer standing on the track perceives the pole as shorter, thus passing faster at a given point compared to what the vaulter would observe within his frame, where the pole remains at its full rest length. This difference underscores the phenomenon of relativity of simultaneity, showing how each observer's time in space can lead to different perceptions of simultaneously occurring events.

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Most popular questions from this chapter

(a) Using the relativistic expression for total energy \(E\) and the magnitude \(p\) of the momentum of a particle, show that the two quantities are related by \(E^{2}=p^{2} c^{2}+\left(m c^{2}\right)^{2} \cdot(b)\) Use this expression to determine the linear momentum of a proton with a kinetic energy of \(1000 \mathrm{MeV}\).

A rocket launched outward from Earth has a speed of 0.100 c relative to Earth. The rocket is directed toward an incoming meteor that may hit the planet. If the meteor moves with a speed of 0.250 c relative to the rocket and directly toward it, what is the velocity of the meteor as observed from Earth?

A speedboat can travel with a speed of \(50 \mathrm{~m} / \mathrm{s}\) in still water. If the boat is in a river that has a flow speed of \(5.0 \mathrm{~m} / \mathrm{s},\) (a) find the maximum and minimum values of the boat's speed relative to an observer on the riverbank. (b) What is the time difference between a downriver trip (with the current) of \(1000 \mathrm{~m}\) and an upriver trip (against the current)?

A spaceship containing an astronaut travels at a speed of \(0.60 c\) relative to a second inertial observer. (a) Who measures proper time intervals in the ship and the proper length of the ship: (1) the astronaut in the ship, (2) the second observer, or (3) neither? (b) How much time does a clock on board the spaceship appear to lose in a day, according to the second observer? (c) If the second observer measures a length of \(110 \mathrm{~m}\) for the ship, what is its "proper" length"? (d) What is the total energy of the astronaut according to the astronaut if her mass is \(70 \mathrm{~kg} ?\) (e) Repeat part (d) from the viewpoint of the second inertial observer.

One of a pair of 25 -year-old twins takes a round trip through space while the other twin remains on Earth. The traveling twin moves at a speed of \(0.95 c\) for a total of 39 years, according to Earth time. Assuming that special relativity applies for the entire trip (that is, neglect accelerations at the start, end, and turnaround), (a) what are the twins' ages when the traveling twin returns to Earth? (b) On the round trip, how far did the traveling twin go according to the traveling twin? (c) On the round trip, how far did the traveling twin go according to the Earth twin? Which of your answers to parts (b) and (c) are proper lengths, if any?
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