Reflection and Mirrors Legacy Problem #9 Guided Solution

Here is a question about a concave spherical mirror with a focal length of 50 centimeters. We have several different object distances and what we wish to calculate is the image distances for the various object distances. This demands that we use the mirror equation. The mirror equation states that the reciprocal of the image distance plus the reciprocal of the object distance is equal to the reciprocal of the focal length. Now as I step you through the solutions to the five questions, I highly recommend that you get out a sheet of paper or scratch paper that you can begin to write some things down. Because if you're uncomfortable with this mathematics, it's going to be a lot easier if you're actually looking at numbers on a sheet of paper. So I begin by taking the equation and rearranging it so that I can get the image distance of that term by itself on the same side of the equation. That demands that I subtract from both sides of the equation the reciprocal of the object distance. So my equation now becomes 1 divided by di equals 1 divided by f minus 1 divided by the object. Now I know that focal length is 50 centimeters so I can substitute that into the equation and for each of the five parts now, I can use the equation 1 over di equals 1 over 50 minus 1 over the object. And then I can begin to substitute in 125 and 100 and 75 and 50 and 25 centimeters into the object distance location in this equation. Now the equation is complicated by the fact that we have reciprocals here in every term. So when I do my substitutions and I solve for the right side of the equation, what I will have is not the answer, but the reciprocal of the answer after all the equation states that 1 over di equals 1 divided by 50 minus 1 divided by the object distance. So for part A, when I do my math, I get the following 1 over di equals 1 over 50 minus 1 over 125. Now when I evaluate the right side of the equation on my calculator, depending on whether it is a graphing calculator or a scientific calculator, I am probably going to get something like 0.0120 as my answer to the right side of the equation. That is 1 over di equals 0.0120 in decibel form. Now that is not the answer, but if I take the reciprocal of 0.0120, I will get the answer. So 1 divided by 0.0120 gives me a di value of 83.3 centimeters. That is the di that corresponds to an object distance of 125 centimeters. Now each part of this problem is going to be done the same way, just substituting different numbers in for dou. So for part B, it would go like this. One divided by 50 minus 1 divided by 100 evaluates on my calculator to be 0.01. Take the reciprocal of 0.01, and I get 100 centimeters as my answer. And again I can repeat for the various numbers that are provided here. Now I want to step you through part D because that is a little more complicated because in that part you are going to use an object distance of 50 centimeters, so the math comes down to this. 1 over di is equal to 1 over 50 minus 1 over 50, and any number subtracted from itself comes out to be 0. So you end up with 1 over di equals 0. Now 0 is not the answer, that is the reciprocal of the answer. So if you wish to find the answer, you have to take the reciprocal of 0. That is you have to go 1 divided by 0, and that will give you the answer. Problem is when you do that on a graphing calculator and many other calculators, you find out that 1 over 1 divided by 0 has no answer. In other words, for the case of the object being at 50 centimeters, the focal point itself, there is no answer to this question, meaning there is no physical image produced when the object is located at the focal point. So we have to just simply say no answer for part D. For part E, it is a little different as well, because for the first time you end up getting a negative answer. That negative answer simply means that when the object is at 25 centimeters, the image is located on the negative side of the mirror, which means behind the mirror.