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I am reading some materials on relativistic effect but since it is quite abstract and I need to think the situation between moving frame and rest frame, it becomes quite confusing. I setup a situation and some questions with answer (from my understand), could you please tell me if my understanding is correct or not?

A moving frame, call primed frame, attached to a moving train at speed U close to speed of light, you are sitting in the train. I am sitting in the platform, call (unprimed) frame.

1) If you measure the length of the train and get [tex]L_0[/tex], in my point of view, the length of train is short than [tex]L_0[/tex], right?

2) You carry a watch with you, you pretty sure the minute hand of the watch move every 60 second. However, in my point of view, I see the minute hand of your watch is moving ahead more than 60 second. Namely, in my point of view, your watch is running slower than mine.

3) The most confusing concept is the energy in relativistic limit. First, generally, I wonder if the total energy of a particle should be written as

[tex]

E = mc^2

[/tex]

??? And this TOTAL energy should include two parts: kinetic energy and rest energy? This is,

[tex]

\textnormal{total energy} = mc^2 = KE + m_0 c^2

[/tex]

There, [tex]m[/tex] is the mass when the particle is in motion

[tex]

m = \frac{m_0}{\sqrt{1-u^2/c^2}}

[/tex]

What confusing me is: when [tex]u\ll c[/tex], then [tex]m=m_0[/tex], the total energy becomes

[tex]

m_0c^2 = KE + m_0 c^2

[/tex]

this gives kinetic energy go to zero? But I think this should give the non-relativistic case. What's going on?

4) Still about energy. The total energy in relativistic limit can also expressed in terms of relativistic momentum

[tex]

\textnormal{Total energy} = mc^2 = \sqrt{p^2c^2 + m_0^2c^4}

[/tex]

Is this result exactly the same as the previous definition (E = KE + m_0c^2) ?

Again, if [tex]u\ll c[/tex], [tex]p \to m_0 v[/tex], then the total energy becomes [tex]m_0c^2[/tex] ??? In this non-relativistic limit, I think total energy should be [tex]\frac{m_0v^2}{2}[/tex]

5) The last question is: in non-relativistic case, the kinetic energy for particle with mass [tex]m=m_0[/tex] be

[tex]KE = \frac{mv^2}{2}[/tex]

For relativistic case,

[tex]KE = \frac{m_0c^2}{\sqrt{1-u^2/c^2}} - \m_0c^2 = \m_0c^2\left(\frac{1}{1-u^2/c^2} - 1\right)[/tex]

If [tex]u=c[/tex], then the first term in the parentheses go to infinity???

All of these questions about energy are really confusing me. Would you please give me some clues what's going on?

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# Confuse about the relativistic effect

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