Fall 2011 Homework 23 Solutions

Fall 2011 Homework 23 Solutions - Physics 21 Fall 2011...

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Physics 21 Fall, 2011 Solution to HW-23 K 20-53 One way to monitor global warming is to measure the average temperature of the ocean. Researchers are doing this by measuring the time it takes sound pulses to travel un- derwater over large distances. At a depth of 1000 m, where ocean temperatures hold steady near 4 C, the average sound speed is 1480 m/s. It’s known from laboratory measurements that the sound speed increases 4.0 m/s for every 1.0 C in- crease in temperature. In one experiment, where sounds generated near California are detected in the South Pacific, the sound waves travel 7700 km. If the smallest time change that can be reliably detected is 1.0 s, what is the smallest change in average temperature that can be measured? Let D be the distance the waves travel and v be their veloc- ity. Then their travel time is t = D/v . dt dv = D v 2 t D v 2 v, where we dropped the minus sign. We can relate the changes in velocity and temperature: v T 4 . 0 m / s , so v (4 . 0 m / s)∆ T. Substituting the expression for ∆ v into the expression above for ∆ t , we obtain t D v 2 (4 . 0 m / s)∆ T. T v 2 D (4 .
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