| ### Statement | | ### Statement |
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| $1.1.7.$ A submarine diving vertically and uniformly emits sound pulses of duration $\tau_0$. The duration of the pulse reflected from the bottom is $\tau$. The speed of sound in water is $c$. At what speed does the submarine dive? | | $1.1.7.$ A submarine diving vertically and uniformly emits sound pulses of duration $\tau_0$. The duration of the pulse reflected from the bottom is $\tau$. The speed of sound in water is $c$. At what speed does the submarine dive? |
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| ### Solution | | ### Solution |
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| Let's consider what the change of signal duration is related to. For this purpose, it is convenient to pass from the signal duration to its length. | | Let's consider what the change of signal duration is related to. For this purpose, it is convenient to pass from the signal duration to its length. |
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| Let the signal emitted by the sonar on the boat have the form shown in the figure, i.e. its intensity jumps from zero to some value at time $t$, remains constant during time $\tau_{0}$ and again jumps to zero at time $t + \tau_{0}$. | | Let the signal emitted by the sonar on the boat have the form shown in the figure, i.e. its intensity jumps from zero to some value at time $t$, remains constant during time $\tau_{0}$ and again jumps to zero at time $t + \tau_{0}$. |
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| If the boat is stationary, the extent of the radiated signal (the distance between the leading and trailing edges) is $l_{0} = V \tau_{0}$. If the boat is sinking, the extent of the signal becomes smaller. | | If the boat is stationary, the extent of the radiated signal (the distance between the leading and trailing edges) is $l_{0} = V \tau_{0}$. If the boat is sinking, the extent of the signal becomes smaller. |
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| Indeed, let $AC = u \tau_{0}$, where $u$ is the submergence velocity of the boat. | | Indeed, let $AC = u \tau_{0}$, where $u$ is the submergence velocity of the boat. |
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| The leading edge of the signal will move to point $B$ in time $\tau_{0}$, having travelled a distance $l_{0}$. | | The leading edge of the signal will move to point $B$ in time $\tau_{0}$, having travelled a distance $l_{0}$. |
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| Hence, the signal length is equal to | | Hence, the signal length is equal to |
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| $$ | | $$ |
| l_{1} = BC = ( V - u ) \tau_{0} | | l_{1} = BC = ( V - u ) \tau_{0} |
| $$ | | $$ |
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| This signal reaches the bottom, discharges from it without changing its extent, and travels towards the boat. | | This signal reaches the bottom, discharges from it without changing its extent, and travels towards the boat. |
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| At some point in time, the leading edge of the signal will align with the boat. | | At some point in time, the leading edge of the signal will align with the boat. |
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| The rear front of the signal continues to move with velocity $V + u$ relative to the boat, therefore, the duration of the signal received on the boat is equal to | | The rear front of the signal continues to move with velocity $V + u$ relative to the boat, therefore, the duration of the signal received on the boat is equal to |
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| $$ | | $$ |
| \tau = \frac{l_{1} }{V + u} = \frac{(V - u) \tau_{0} }{V + u} | | \tau = \frac{l_{1} }{V + u} = \frac{(V - u) \tau_{0} }{V + u} |
| $$ | | $$ |
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| From here | | From here |
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| $$ | | $$ |
| u = V \frac{ \tau_{0} - \tau}{ \tau_{0} + \tau} | | u = V \frac{ \tau_{0} - \tau}{ \tau_{0} + \tau} |
| $$ | | $$ |
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