# Length contraction numerical problem

• Pushoam
In summary, the conversation discusses the calculation of kinetic energy and speed of a proton using both non-relativistic and relativistic equations. It is determined that in order to accurately calculate the speed of the proton, its kinetic energy must be taken as 1019 eV. The conversation also mentions the time it would take for the proton to cross the galaxy in both the galaxy frame and the proton frame. The final output is a suggestion to convert the time taken in the proton frame to seconds.
Pushoam

## Homework Equations

1) k.e. = (1/2)mv2, non- relatvistic

2)K.E. = m0 c2 (ϒ -1), relativistic

3) L=L0/ϒ , Length contraction

## The Attempt at a Solution

(A) Taking 1019 eV to be the kinetic energy of the proton,
Non- relativistic calculation
(1/2)mv2 = 1019 eV = 1.6

v = √[ (3.2/1.67)* 1027 ] = 4.3 * 1013 m/s

How is kinetic energy defined relativistically?

K.E. = m0 c2 (ϒ -1)

We can't take 1019 eV as total energy of the proton as if we take so then we won't be able to calculate speed of the proton.

So, for calculating speed of the proton, I have to take 1019 eV as the kinetic energy of proton.

So,
ϒ2 = [ {1019 eV/ m0 c2}+1]2

m0 c2 = 939 MeV

Solving it gives,
1 - v2/c2 = [939 * 10-13]2

v ≈ c

Hence, w.r.t. galaxy frame the proton will take 105 years to cross the galaxy.
B)

W.r.t. the proton frame,
the length of the galaxy is shortened to 105 [939 * 10-13] light years.Hence, the time taken = 105 [939 * 10-13] yearsIs this correct?

Pushoam said:
Hence, the time taken = 105 [939 * 10-13] years
Hi Pushoam:

Your work looks OK to me. However, I suggest converting the (B) answer to seconds.

Regards,
Buzz

## 1. What is length contraction and how does it work?

Length contraction is a phenomenon in which an object's length appears to be shortened when observed from a different frame of reference that is moving at a high velocity. This is a consequence of Einstein's theory of relativity, in which the concept of space and time are relative to the observer's frame of reference. The amount of contraction is calculated using the Lorentz factor, which takes into account the object's velocity.

## 2. How is length contraction related to time dilation?

Length contraction and time dilation are both consequences of the theory of relativity. Time dilation refers to the slowing down of time for an object in motion, while length contraction refers to the shortening of an object's length. These two phenomena are interconnected and occur simultaneously in order to maintain the speed of light as a constant in all frames of reference.

## 3. Can you provide an example of a length contraction numerical problem?

One example of a length contraction numerical problem is the famous "pole in a barn" thought experiment. In this scenario, a pole of a certain length is moving at a high velocity towards a barn. Observers inside the barn measure the length of the pole to be shorter than its actual length due to length contraction. This can be calculated using the Lorentz factor and the velocity of the pole.

## 4. Does length contraction only occur at high velocities?

Yes, length contraction is only noticeable at high velocities that are a significant fraction of the speed of light. This is because the Lorentz factor becomes significant at these speeds, leading to a noticeable contraction of length. At everyday speeds, the effects of length contraction are negligible and not noticeable.

## 5. How does length contraction affect objects in everyday life?

In everyday life, the effects of length contraction are negligible and not noticeable. However, in highly sensitive scientific experiments, such as particle accelerators, the effects of length contraction must be taken into account in order to make accurate measurements. Additionally, GPS systems must also take into account the effects of length contraction and time dilation in order to function accurately.

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