Special relativity question.

In summary, the object's total energy must be less than its rest energy for it to be traveling at a speed less than its resting energy. However, this answer is complicated and requires definitions for momentum, rest energy, and total energy that are beyond the scope of this question.
  • #1
tkav1980
47
1
Ok here's the short story. A friend of mine is in grad school for physics. I have a BS in physics, however i graduated in 2002 and haven't used anything i learned as i don't work in the field. I always try to think up questions to stump him. This time he got me. Here's what he asked me. In special relativity, At what speed does an object need to travel for its Total energy to be less than its resting energy. This one is way out of my league at this point.

If i posted this in the wrong place i apologige.

p.s. i asked him about the object having zero mass and didnt get a response.
 
Physics news on Phys.org
  • #2
tkav1980 said:
In special relativity, At what speed does an object need to travel for its Total energy to be less than its resting energy

Impossible.
[tex]E^2=m^2c^4 +p^2 c^2=E_0 ^2 + (pc)^2 \implies E \geq E_0[/tex]
 
  • #3
Nabeshin said:
Impossible.
[tex][/tex]

But i don't see a solution for velocity? Even if that number was a negative number wouldn't there be at least a value that could be assigned. Sorry if I am annoying I've been at work a long time at this point in my day. The brain shut off, but from my understanding of what he asked the total energy of the particle ( the rest energy plus the kenetic energy) must be less than it's rest energy. wouldn't there be a solution in terms of velocity. at least a way to mathmatically solve for it weather the answer is real or imaginary?
 
Last edited:
  • #4
tkav1980 said:
Ok here's the short story. A friend of mine is in grad school for physics. I have a BS in physics, however i graduated in 2002 and haven't used anything i learned as i don't work in the field. I always try to think up questions to stump him. This time he got me. Here's what he asked me. In special relativity, At what speed does an object need to travel for its Total energy to be less than its resting energy. This one is way out of my league at this point.

If i posted this in the wrong place i apologige.

p.s. i asked him about the object having zero mass and didnt get a response.

[tex]TE=\frac{m_0c^2}{\sqrt{1-(v/c)^2}}[/tex]
[tex]RE=m_0c^2[/tex]

Find [tex]v[/tex] such that [tex]TE<RE[/tex] :-)
 
  • #5
tkav1980 said:
But i don't see a solution for velocity? Even if that number was a negative number wouldn't there be at least a value that could be assigned. Sorry if I am annoying I've been at work a long time at this point in my day. The brain shut off, but from my understanding of what he asked the total energy of the particle ( the rest energy plus the kenetic energy) must be less than it's rest energy. wouldn't there be a solution in terms of velocity. at least a way to mathmatically solve for it weather the answer is real or imaginary?
In units such that c=1, we have [itex]E=\gamma m -m[/itex], where E is the kinetic energy, [itex]\gamma m[/itex] is the total energy, and m is the rest energy. You're looking for a velocity v such that [itex]\gamma m< m[/itex]. This is equivalent to [itex]\gamma<1[/itex], but

[tex]\gamma=\frac{1}{\sqrt{1-v^2}}>1[/tex]

However, all of the above is for massive particles...and by that I mean particles with mass m>0.

Every particle satisfies an equation of the form [itex]-E^2+p^2=A[/itex]. If [itex]A\leq 0[/itex], we write [itex]A=-m^2[/itex], where m is defined to be >0, and is called the "mass" of the particle. If we insist on writing [itex]A=-m^2[/itex] even when A>0 (which would be appropriate if we intend to call m the "mass"), m must be imaginary. These particles are called tachyons.

It's a bit more convenient to write [itex]A=n^2[/itex], where n=im>0. The equation that we would write as [itex]E^2=p^2+m^2[/itex] for massive particles would be written as [itex]E^2=p^2-n^2[/itex]. I haven't really thought about how to define momentum, rest energy or total energy for tachyons, but it's clear that these are the things you need to work out if you're going to really answer the question.

The really short answer is of course v>c (if it's possible at all...I haven't verified that it is), but the answer isn't complete without the appropriate definitions.
 

1. What is special relativity?

Special relativity is a theory developed by Albert Einstein in 1905 that describes the relationship between space and time. It states that the laws of physics are the same for all observers in uniform motion, and that the speed of light in a vacuum is constant for all observers.

2. How does special relativity differ from classical mechanics?

While classical mechanics assumes that time and space are absolute, special relativity shows that they are actually relative and can change depending on the observer's frame of reference. Special relativity also introduces the concept of time dilation and length contraction, which do not exist in classical mechanics.

3. What is the equation for time dilation in special relativity?

The equation for time dilation is t' = t / √(1 - v^2/c^2), where t' is the time measured by an observer in motion, t is the time measured by a stationary observer, v is the relative velocity between the two observers, and c is the speed of light in a vacuum.

4. Can special relativity be applied to everyday situations?

Yes, special relativity has been experimentally proven and is used in various technologies such as GPS systems, particle accelerators, and nuclear reactors. However, its effects are only noticeable at speeds close to the speed of light, so it may not be apparent in everyday situations.

5. How does special relativity relate to general relativity?

Special relativity only applies to objects moving at constant velocities in a straight line, while general relativity applies to all types of motion, including acceleration and gravity. General relativity is a more comprehensive theory that builds upon the principles of special relativity to describe the relationship between space, time, and gravity.

Similar threads

  • Special and General Relativity
Replies
15
Views
1K
  • Special and General Relativity
Replies
15
Views
715
  • Special and General Relativity
Replies
10
Views
670
Replies
32
Views
836
  • Special and General Relativity
Replies
11
Views
925
  • Special and General Relativity
Replies
10
Views
1K
  • Special and General Relativity
Replies
7
Views
793
  • Special and General Relativity
Replies
11
Views
1K
  • Special and General Relativity
Replies
17
Views
390
Back
Top