Time taken for sound to travel between two moving observers

[Saptarshi Sarkar]

Homework Statement

A source S of frequency $f_0$ and an observer O, moving with speeds $v_1$ and $v_2$ respectively, are moving away from each other. When they are separated by distance a (t=0), a sound pulse is emitted by the source. Suppose velocity of sound to be $v_s$ and calculate the time $t_1$ that it takes for the pulse to be received by O.

Relevant Equations

Total distance the pulse needs to travel:
$D = a + v_1t_1$

Speed of sound pulse = $v_s - v_2$

So,
$t_1 = \frac {a + v_1t_1} {v_s - v_2}$

But the solution should be

$t_1 = \frac a {v_s - v_2}$

I assumed the following -

1. I did not consider the frequency as the Doppler shift in frequency was not asked.

2. I did not add the distance the source moved in time $t_1$ to the total distance traveled by the wave as the pulse was emitted at t=0.

Is any of my assumptions wrong?

 

[haruspex]

Please explain your reasoning for your expression for D.
You could sketch the motions of source, observer and pulse on a distance-time graph.

 

[PeroK]

So,
$t_1 = \frac {a + v_1t_1} {v_s - v_2}$

I assumed the following -

2. I did not add the distance the source moved in time $t_1$ to the total distance traveled by the wave as the pulse was emitted at t=0.

How is that equation based on your assumption?

Are taking $v_2$ to be the speed of the source?

 

[Saptarshi Sarkar]

Sorry, I guess I messed up the velocities. I will edit the question and add a sketch tomorrow morning.

 

[Saptarshi Sarkar]

Can't edit the question, so posting it here

Homework Statement::

A source S of frequency $f_0$ and an observer O, moving with speeds $v_1$ and $v_2$ respectively, are moving away from each other. When they are separated by distance a (t=0), a sound pulse is emitted by the source. Suppose velocity of sound to be $v_s$ and calculate the time $t_1$ that it takes for the pulse to be received by O.
Homework Equations::

Total distance the pulse needs to travel:
$D = a + v_2t_1$

Speed of sound pulse = $v_s - v_1$

So,
$t_1 = \frac {a + v_2t_1} {v_s - v_1}$

But the solution should be

$t_1 = \frac a {v_s - v_2}$

I assumed the following -

1. I did not consider the frequency as the Doppler shift in frequency was not asked.

2. I did not add the distance the source moved in time $t_1$ to the total distance traveled by the wave as the pulse was emitted at t=0.

Is any of my assumptions wrong?

 

[PeroK]

The speed of sound in a medium is independent of the motion of the source.

 

[Saptarshi Sarkar]

The speed of sound in a medium is independent of the motion of the source.

Thanks!