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Monday, June 1, 2015

Problem of the Week

Good morning from the Center of Math, and welcome to June! It's a rainy morning here in Cambridge. It's a coincidence that our Problem of the Week (which we posted to Facebook, Twitter, and Google+) involves water. It's slightly deceptive, so be careful when reading it:
Click on any picture to expand!
Ok, so this problem has a little more physics than our usual problem of the week. But it's nothing we can't handle. Take some time to try it yourself! My solution is below:

I started by drawing the well. I labeled what we will be measuring: the height h from the top of the well to the water level. We also have some information already: we know that it takes 3 total seconds from releasing the stone and the sound of the splash hitting your ears. I labeled the drop from the top of the well to the water as time t, and I gave time (3-t) to the soundwave that travels back up the well.

Then, in the image directly below, I made sure to write out all of the terms that I will be using in my solution. I restated the given values from the problem, and I wrote out my time divisions again. Then, I started to determine how to find my height. 

I used two equations set equal to h. The first measures the distance that the stone falls from the lip of the well to the water below. This called for a simple kinematics equation, that measures projectile motion in the vertical direction. Our initial velocity was 0 m/s, so this equation reduced to involving just the acceleration due to gravity as the stone fell.

The other equation for height comes from the distance the soundwave travels up after the splash. In this case, the height h is equal to the speed of sound multiplied by the time it took to travel.

We can see what the two equations for height reduced to directly below, and then I set the two equations equal to each other and solved for t.

The next step involved the quadratic equation! I only needed the positive solution, because you can't have a negative passage of time in this scenario, so I didn't bother finding the second solution.

Then, I plugged the value I found for t back into my equations for height. I found that the well is approximately 41 meters deep.

Did you solve this problem another way? Do you have any questions about my method? Let me know in the comments!

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