Question

A train of length $100m$ is moving in a hilly region. At what speed must it approach a tunnel of length $80m$ so that a person 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

Answer: (D)

$0.6c$

Length contraction is given by

$L=L_0\sqrt{1-\frac{v^2}{c^2}}$

where $L=$contraction in length

$L_0=$length of moving object in rest

$v=$ relative velocity of object with respect to observer

$c=$ velocity of light

now given , $L=80m$

$L_0=100m$

$v=v\pm 0$ as the observer is at rest

therefore ,

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

or $\frac{4}{5}=\sqrt{1-v^2/c^2}$

or $1-\frac{v^2}{c^2}=\frac{16}{25}$

or $\frac{v^2}{c^2}=1-\frac{16}{25}=\frac{9}{25}$

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