A relativistic train with a proper length 80 m passes through a tunnel with a pr

Question

A relativistic train with a proper length 80 m passes through a tunnel with a proper length 75 m. (a) If the train’s speed is 0.45c, is the train every completely inside the tunnel? (b) At this speed, how much time elapses for a cross-country trip of 5000 km, as measured on the train? ANSWER IN THE FOLLOWING FORMAT:

Key Idea:

Write down a short statement of the physical principle which you think is the most

relevant to solving the problem. Describe the elements of the problem which indicated that this

physical principle might be applicable.

Stock of Data:

Make a list of the numerical data you are given in the problem description. Be

sure to include

units. List quantities you are asked to calculate with trailing question marks.

This step may include a diagram indicating your understanding of the problem.

I.D. Equation:

Choose an equation which reflects both the key idea of your problem and

information from your stock of data. You may need more than one equation.

Solve:

If the equation or equations you have chosen are not already solved for you unknown

then do the algebra to accomplish this now. Be sure to show enough detail in each step so that

a reader can follow your work.

Substitute in Numerical Data:

Once you have an analytic solution, substitute numerical

values into the analytic solution to obtain the final answer.

Sanity Check:

Answer the question. “Does this answer make sense in light of the rest of the

knowledge I possess?” The sanity check could include a back of the envelope calculation done

with a single significant digit or a unit analysis of the final answer.

Explanation / Answer

5,000 km
0.45c = 135000 km/s

Time = Distance/Speed
Time = 5,000/135000
Time = 0.03704 s

But thats the proper time, we need to figure out the dilated time, so lets pulg this into the equation:
T = t x sqrt(1-v^2/c^2)
T is the dilated time
t is the proper time, 0.03704 s
v is 0.45
c = 1

T = 0.03704 x sqrt(1 - 0.45^2/1^2)
T = 0.03308 s

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