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
• B. 0.8c
• C. 0.64c
• D. 0.6c
• E. 0.36c

Answer 1

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

$l=l_0\sqrt{1-v^2/c^2}$

where $l=$ contracted length in moving position,

$l_0=$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 $l_{0=}100m$ , $l=80m$ ,

therefore by above relation ,

$80=100\sqrt{1-v^2/c^2}$

or $\sqrt{1-v^2/c^2}=80/100$

or $1-v^2/c^2=(4/5{)}^2$

or $v^2/c^2=1-16/25=9/25$

or $v^2=9c^2/25$

or $v=3c/5=0.6c$