Wave Race 2: Explain the logic behind this formulary?

[pugfug90]

https://www.physicsforums.com/showthread.php?t=160610
Continuing off this problem..
My teacher wrote the solution on the board (and conveniently went to the hospital before our period).
Problem again..

Homework Statement

The velocity of the transverse waves produced by an earthquake is 8.9km/s while that of the longitudinal waves is 5.1km/s. A seismograph records the arrival of the transverse waves 73s before that of the longitudinal waves. How far away was the earthquakes?

The Attempt at a Solution

Doc Al's formula was.. the distance/displacement of transverse velocity is equal to the distance/displacement of longitudinal velocity * time + 73s..
Or..
d=vt=(8.9km/s)* T=(5.1km/s) * (T+73s)
And eventually, the T will get isolated..
================================
On the board..
His formula was
Time_transverse=[V_longitudinalΔt]/(v_transverse-V_longitudinal)

Δt being 73s, the rest being self explanatory.

And it turns out.. seconds is also 98s! Seems like fiddling of numbers until eventually getting the right answer.. but nah

And how in the world did he derive that formula.. Time of transverse wave is equal to the velocity of the longitudinal wave times the change in time divided by the difference in velocity.

 

[G01]

Both waves travel a distance of d.

This involves a little algebra. I'll start the derivation and let you finish it. (It's good practice!)

Consider the following and then see if you can finish what I start and derive his formula.

Transverse Wave:

$$d=V_Tt$$

Longitudinal Wave:

$$d = V_L(t+ \Delta t)$$

Since both waves travel the same distances, we can eliminate d from the problem as follows:

$$V_Tt = d = V_L(t+ \Delta t)$$
$$V_Tt = V_L(t+ \Delta t)$$

Now from here on you should be able to isolate t and solve for it. You should get your teachers formula. Good Luck! If you need any more help, don't hesitate to ask.

G01

 

[pugfug90]

Ahha..
So V_transverset=V_longitudinalt + V_longitudinalΔt

Then..

V_transverset - V_longitudinalt = V_longitudinalΔt

Then..

t(V_transverse - V_longitudinal) =V_longitudinalΔt

Then..

t=[V_longitudinalΔt]/(V_transverse - V_longitudinal)

Which is what teacher got. I thought he found some newer innovative way to derive that.. If he had showed how he derived the formula, it would've helped. Thanks

 

[G01]

Nice Job. Its no problem. Glad I could help!