Length Contraction Formula

Einstein’s theory of special relativity states that “The length of objects moving at relativistic speeds undergoes a contraction along the dimension of motion”. An observer at rest (in relation to the moving object) would perceive the moving object to be shorter. As perceived from a stationary or at rest observer reference frame, the object length is really contracted. The object’s length contraction is solely determined by its velocity in relation to the observer. The concept of length contraction and its formula is discussed below.

Length Contraction

When an object travels at the speed of light, length contraction is considered. As a result, relativity enters the scene. As a result, when an item travels at the speed of light, it experiences length contraction. If a body is travelling at the speed of light and is linked to an observer, this is expressed as a decrease in length. Length contraction occurs only in the direction in which the body is travelling. 

It is the occurrence or phenomenon; that a moving object’s length when measured is found shorter than its proper length, which is the length as measured in the object’s own rest frame. This contraction phenomenon was postulated by George FitzGerald (1889) and Hendrik Antoon Lorentz (1892).

But in 1905, Albert Einstein was the first to demonstrate that this contraction did not require motion through a supposed ether, but could be explained using special relativity theory. This theory changed all our notions of space, time, and simultaneity. A very important point to remember is that length contraction cannot be measured in the object’s rest frame, but only in a frame in which the observed object is in motion.

Formula

According to Einstein’s theory of special relativity, the distance between two points can differ in different reference frames. An object we’re measuring will be at rest in one reference frame. This is the correct length, and it is referred to as .

or

L = 

where, 

• L is the length of an object with is in relativistic speed
• L₀ is the length of an object at rest
• c is the velocity of light 
• v is the velocity of the object
• γ = 

Sample Problems

Question 1. An object of length 20 m travels at a velocity of 0.65c. Calculate the length contraction.

Solution:

Given: L₀ = 20 m, v = 0.65 c

The length contraction formula states that: L = 

Substitute the given values in the formula above. Then,

L =  

= 20 × √1- (0.65)²

L = 15.198 m

Question 2. How can you say that c is the limiting speed in universe using time dilation and length contraction formulas?

Solution:

The length contraction formula states that: L = .

The time dilation formula states that: 

If something moves at the speed of light, the length becomes 0 and the time becomes infinity. So, no signal or interaction can move faster than the speed of light because they will simply not be able to occur.

Question 3. How fast would a 6.0 m-long sports car have to be going past you in order for it to appear only 5.5 m long?

Solution:

The length contraction formula can be written as: L = 

where γ = 

Given: L₀ = 6 m, L = 5.5 m

Substitute these values in the above formula. Then,

Question 4. Relativistic effects such as time dilation and length contraction are present for cars and airplanes. Why do these effects seem strange to us?

Solution:

The speed of cars and airplanes is much less then the speed of light. And the realistic effects like time dilation and length contraction are most observable when the moving observer is moving with a speed that is very near to the speed of the light.

Hence such effects are almost miniscule for cars and airplanes as their speed is in comparable to that of light, hence they seem strange to us.

Question 5. An object of length 36 m travels at a velocity of 0.45 c. Calculate the length contraction.

Solution:

Given: L0 = 36 m, v = 0.45 c

The length contraction formula states that: 

Substitute the given values in the formula above. Then,

L = 

= 36 × √1- (0.45)²

L = 32.149 m