A sand scorpion can detect the motion of a nearby beetle (its prey) by the waves

Question

A sand scorpion can detect the motion of a nearby beetle (its prey) by the waves the motion sends along the sand surface (Fig. below). The waves are of two types: transverse waves traveling at vt = 56 m/s and longitudinal waves traveling at vl = 155 m/s . If a sudden motion sends out such waves, a scorpion can tell the distance of the beetle from the difference ?t in the arrival times of the waves at its leg nearest the beetle. If ?t = 5.3 ms, what is the beetle's distance (in cm)?

PLEASE BE SPECIFIC

Explanation / Answer

I can't access the pic, but I think there is enough info to help.

the time it takes for the transverse waves to arrive is

time(t) = distance/speed = distance/58m/s

the time for long. waves to arrive is

time(L) = distance/speed = distance/165m/s

the distance is the same in both cases...so we have

time(L) - time(t) = 0.0053s, so

0.0053s = D/58 - D/165 = D(1/58-1/165)

D = 0.0053/[1/58-1/165] = 0.0053/0.011 = 0.474 m
D = 47.4 cm.....Beetle's distance

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