On a warm summer day $(31 ^\circ \textrm{C})$, it takes 4.80 s for an echo to return from a cliff across a lake. On a winter day, it takes 5.20 s. What is the temperature on the winter day?

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. Speed of sound is 331 plus 0.6 times the temperature in Celsius. And we know that distance is speed times time. And the distance across the lake is the same in the summer and the winter. So, the distance is the speed of sound in the summer times the time it takes to do that round trip across the lake and back in the summer, and that's gonna be the same as the speed of sound in the winter times the time it takes to do the same round trip in the winter. So, we can put these 2 parts equal to each other. Here with substitutions for speeds of sound in place of vs and vw. So, for the summer I have 331 plus 0.6 times temperature in the summer. And then for v winter is 331 plus 0.6 times temperature in the winter. And then I switch the sides around, multiply through by this time in the winter, 331 times tw plus 0.6 times tw times temperature in the winter. And this is the summer side unchanged but just move to the right hand side. And then subtract away 331 tw from both sides and you're left with 0.6 time winter temperature winter equals all this stuff here. And then divide both sides by 0.6 times time in the winter, and you get temperature in the winter isolated. And that's gonna be 331 plus 0.6 times temperature in the summer, all times the time taken in the summer minus 331 times time in the winter divided by 0.6 time in the winter. And this gives 331 plus 0.6 times 31 degrees Celsius times 4.8 seconds for the round trip in the summer minus 331 times 5.2 seconds in the winter all divided by 0.6 times 5.2, which gives about negative 14 degrees Celsius must have been the temperature in the winter. So, you can measure temperature by measuring the time it takes for sound to travel.