Calculating Wave Speed & Distance from Boat

[Resmo112]

Homework Statement

the speed of waves on a lake depends on frequency. For waves of frequency 1.0 Hz the wave speed is 1.56 m/s; for 2.0 Hz waves the speed is .78 m/s. the 2.0 Hz waves reach you 120 seconds after the 1.0Hz waves generated by the same boat. How far away is the boat.

Homework Equations

we have 2 velocities and 2 frequencies the only equation I can find that includes both of those variables is
v=Lambda *f

The Attempt at a Solution

I'm studying for my final, I'd THINk you could just times the time by the velocity and that'd give you the distance between the the person and the boat. but the problem is it's the DIFFERENCE between 2 waves, that's confusing me, if you multiply 1.56 m/s * 120 s you end up with 187.2 which is really close to the actual answer which is 190m. any help would be greatly appreciated.

 

[olivermsun]

The reason you notice the funny fact that 1.56 m/s * 120 s = the answer is because the first wave travels twice as fast as the second wave.

One systematic way to set up the problem uses only the basic fact that x = vt. Since they've given you the wave velocities, you don't need to do anything with the frequency, wavelength, or any other properties of waves. All you need to know is that they are moving objects with known velocities.

Let x be the position of the boat.
v1 = 1.56 m/s the velocity of 1.0 Hz waves.
v2 = 0.78 m/s the velocity of 2.0 Hz waves.

t1 = the time for the 1.0 Hz waves to reach you.
t2 = t1+120 s the time for the 2.0 Hz waves to reach you.
x = v1 * t1 = v2 * t2
= v2 (t1 + 120 s)

and now solve for t1 and you will see the funny fact emerge.

 

[eczeno]

hi,

so, we have two waves moving from point A to point B, so they go the same distance, let's call that d. but, they are going at different speeds, so they take different times. we don't know those times, but we know the difference between them, so would could say that t_{fast}=t_{slow}-120. now just apply d=v t to both waves.

cheers.

 

[kuruman]

Write two separate expressions for the time it takes each wave to reach you, one for each frequency. Take the difference of the two times and see what you get.

 

[Resmo112]

I get 120 when I do that. I set it up like this

V1*t1=V2*t2 and since T2= T1+120

V1*T1=V2*(t1+120)

V1=1.56 m/s
V2 = .78 ms so

1.56m/s *t1 = .78m/s*(t1+120)
factor the .78 through

1.56*t1 = .78t1*93.6

subtract the .78from 1.56

.78T1 = 93.6 ohhhh wait that's T1 so T1 = 120 t2 + 240 so the total time it takes for the wave to get to me! is 240 seconds so T2 * .78 = the total distance.

Thank you guys so much