Question

An earthquake generates both transverse $(S)$ and longitudinal $(P)$ sound waves in the earth. The speed of $(S)$ waves is about $4.5\, km / s$ and that of $(P)$ waves is about $8.0\, km / s$. A seismograph records $P$ and $S$ waves from an earthquake. The first $P$ wave arrives $4.0 \min$ before the first $S$ wave. The epicentre of the earthquake is located at a distance of about

• A. $25\, km$
• B. $250 \, km$
• C. $2500\, km$
• D. $5000\, km$

Answer 1

Answer: (C)

Distance travelled by both the waves is same.
Let the time taken by the $S$ and $P$ waves to reach the seismograph be $t_1$ and $t_2$ then
$t_1-t_2=4 \min$
Since, $60\, s=1 \min$
$\therefore t_1-t_2=60 \times 4=240\, s$
Let distance of epicentre be $s$. Then
$ s=v_1 t_1=v_2 t_2$
$\Rightarrow 4.5 \times t_1=8 t_2$
$\Rightarrow t_2=\frac{4.5}{8} t_1 $
$\therefore t_1-t_2=240 $
$\Rightarrow t_1\left(1-\frac{4.5}{8}\right)=240 $
$\Rightarrow t_1=\frac{240 \times 8}{3.5}=548.58\, s$
$\therefore s=v_1 t_1=4.5 \times 548.5=2500\, km$