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Time Dilation Formula

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According to the theory of relativity, time dilation is defined as the difference between the elapsed time of two occurrences assessed by either moving relative to each other or by gravitational mass or masses located at different locations.

Consider a clock that is watched by two people. One observer is stationary, while the other is travelling at the speed of light. The existence of time difference between the two clocks is known as time dilation.

What is Time Dilation?

Time dilation is the phenomenon in which two bodies moving relative to each other (or even just a different intensity of gravitational field from each other) experience different rates of time flow.

It refers to a unique situation in which time can pass at different rates in different reference frames. It also depends on the relative velocity of one reference frame to another.

In layman’s terms, time dilation is the measurement of elapsed time using two clocks. In addition, the appropriate time (one-position time) and observer time are two reference frames (two-position time). Furthermore, they are intertwined, and we can determine the time dilation of one by knowing the velocity and speed of the others.

Formula for Time Dilation

The time dilation formula is given by,

T =T0 /√1−(v2/c2)

where,

T is the time observed

T0 is the time observed at rest v is the velocity of the object

c is the velocity of light in a vacuum (3 × 108 m/s2)

Derivation of Time Dilation

To compare the time measurements in the two inertial frames quantitatively, we can link the distances into each other, then quantify each distance in terms of the pulse’s time of travel in the associated reference frame. The resulting equation can then be solved for T in terms of T0 

The lengths D and L are the hypotenuse s of a right triangle. The Pythagorean theorem states that

s2 = D2 + L2

The distances travelled by the light pulse and the spacecraft in time  in the earthbound observer’s frame are 2s and 2L, respectively. In the astronaut’s frame, the length D is the distance travelled by the light pulse in time T0. This gives us three equations to work with:

2s = cT; 2L = vT; 2D = cT0

In both inertial frames, we exploited Einstein’s second postulate by taking the speed of light to be c. We can now plug these results into the Pythagorean theorem’s prior expression:

s2 = D2+ L2

(c × T/2)2  = (c × T0/2)2 + (v × T/2)2

Then we rearrange to obtain

(c × T)2 – (v × T)2 = (c × T0)2

Finally, solving for T in terms of T0 gives us

T =T0 /√1−(v/c)2

This is equivalent to

T = γT0,

where  Î³ is the relativistic factor (often called the Lorentz factor) given by

γ =1/√1−(v2/c2)

and v and c are the speeds of the moving observer and light, respectively.

Sample Problems

Problem 1: Determine the relativistic time, if T0 is 7 years and the velocity of the object is 0.55c.

Solution:

Given:

T0 = 7 years

v = 0.55c

The Formula for time dilation is given by,

T =T0 /√1−(v2/c2)

T = 7/√1-(0.55)2(32 x 1016)/32 x 1016 

T = 7/√1- (0.55)2

T=7/0.8351

T = 8.38 years

Problem 2: What is γ? If v=0.650c.

Solution:

γ = 1/√1−v2/c2

   =1/√1−(0.650c)/c2

   = 1.32

Problem 3: A particle travels at 1.90×108m/s  and lives 2.1×108s  when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

Solution:

Δt = Δτ/√1−v2/c2

    = 2.10×10−8s/√1−(1.90×108m/s)2/(3×108m/s)2 

    = 2.71×10−8s

Problem 4: How does time change over 10 years travelling at a speed of 50% of that of light?

Solution:

T0 =T x√1−(v2/c2)

    = 10 years x √1 – 502/1002

    =10 years x √1 – 2500/10000

    = 10years x √1 – 0.25

    = 10years x √0.75

    = 10years x 0.866

T0= 8.66 years

Problem 5: Given v = 0.95c, T0 = 10 years. Find T which is the time that the earth bound brother measures?

Solution:

T = 10/√(1- (0.95c)2/c2)

T= 10/√(1- 0.952)

T = 10/ 0.312

T = 32 years


Last Updated : 01 Feb, 2022
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