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Time Dilation, Length Contraction, & Relative Speed
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Time Dilation, Length Contraction, Relative Speed are some of the most interesting and puzzling phenomena that puzzled scientists for quite a time. They are not only fun to learn but totally perplexing as well. In this article, we will learn about Time Dilation, Length Contraction, and Relative Speed. All three phenomena are related to Einstein’s Theory of Relativity.
Time Dilation
According to Classical Physics, time is an absolute quantity. But according to the Special Theory of Relativity, time is not an absolute quantity. It depends upon the motion of the frame of reference as we can see in Lorentz’s transformation. If the interval of time (say ticking of a clock) between two signals in an inertial frame S be t, then the time interval between these very two signals in another inertial frame S’ moving with respect to the first will be given by:
This shows that t’ has increased or dilated due to motion, in other words, the clock goes slow according to for S frame with respect to S’
Based on this there are many modern applications for this principle. e.g., a Navigation system, study of subatomic particles and their mean lifetime, etc.
Here,
t = proper time according to the rest frame
t’ = dilated time measured be observer moving with speed v
v = velocity of S frame
Example
Q: A meson has a speed 0.6c with respect to ground and its time of flight with respect to in the rest frame is
A: Given-
Proper time of flight, t =
Speed of the meson, v =0.6c Now according to the equation of time dilation
Also, learn about Lorentz Transformation
Length Contraction
If we measure the length of anything moving relative to our frame, we find its length L to be smaller than the proper length
- Thus, according to Einstein’s Theory of relativity, length contraction is the phenomenon in which the length of an object is measured to be shorter than its proper length measured in the rest frame.
- It is also known as Lorentz contraction and mainly noticeable when the inertial frame of reference moves with a substantial fraction of the speed of light.
- If
is the rest length of the square the length measured by a stationary frame of reference then the length of the square measured in the frame moving with the velocity v is given as
Here,
L= proper length
L’= Contracted length measured in S’ framed with respect to S
v = velocity of S frame
Example
Q: An arrow measures 1 m in length at rest. It is launched at a relativistic velocity of 80% of the velocity of light. What is the length of the arrow due to Lorentz-Fitzgerald contraction as measured by an observer, to whom the arrow is in motion?
A: Given – Length of the arrow, L = 1 m and v = 0.8c
The length of the arrow due to Lorentz-Fitzgerald contraction as measured by an observer is
Relative Speed
The relative speed of object A with respect to another object B is the speed that object A would appear to have to an observer situated on object B moving along with it.
At small speeds of our daily life, we use this formula,
But according to Albert Einstein, the speed of light is constant. Always. No exceptions. It means nothing how fast you are moving relative to a photon, or even if you’re moving at the speed of light itself.
Now to understand this, consider
So, if a particle has a velocity u’ in a frame of reference S’ which is moving with velocity v relative to another frame of reference S, the velocity of the particle in the frame S will be:
Special case:
If the velocity component is along y or direction in the S frame the above equation can be modified as
We hope the above article guides you to understand the important concept of Time Dilation, Length Contraction, Relative Speed. Do practice it now on the Testbook App through the free mock tests.
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Time Dilation Length Contraction Relative Speed FAQs
What is Lorentz Transformation?
The Lorentz transformation equation was first given by H.A Lorentz in the year 1890, which provided the mathematical relation between space and time in one inertial frame of reference with respect to the other. It was accepted widely when Einstein used his assumption for deriving the special theory of relativity
What is Time Dilation?
According to the theory of relativity, time dilation is the difference in the elapsed time as measured by two clocks in different inertial frames.
What is Length Contraction?
According to Einstein’s Theory of relativity, length contraction is the phenomenon in which the length of an object is measured to be shorter than its proper length measured in the rest frame.
When is Lorentz Contraction visible?
Lorentz contraction and mainly noticeable when the inertial frame of reference moves with a substantial fraction of the speed of light.
What is Relative Velocity?
The relative velocity of object A with respect to another object B is the velocity that object A would appear to have to an observer situated on object B moving along with it.
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