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Question

A train at rest has a length of 100 m. At what speed must it approach a tunnel of length 80 m so that an observer at rest with respect to the tunnel will see that the entire train is in the tunnel at one time?

A
1.25c
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B
0.8c
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C
0.64c
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D
0.6c
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E
0.36c
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Solution

The correct option is D 0.6c
At relativistic speeds (speeds comparable to the speed of light) length of a body is contracted , which is given by the following relation
l=l01v2/c2
where l= contracted length in moving position,
l0=length in rest position of body ,
c= velocity of light ,
v= velocity of body ,
to see the whole train in tunnel its length must be contacted atleast to 80m ,
as given l0=100m , l=80m ,
therefore by above relation ,
80=1001v2/c2
or 1v2/c2=80/100
or 1v2/c2=(4/5)2
or v2/c2=116/25=9/25
or v2=9c2/25
or v=3c/5=0.6c

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