**Mirror, mirror on the wall, where is my reflection?**

Is there a linear relationship between the object distance and the image distance for a mirror? Is there a linear relationship between the depth of an image below the surface of water and the actual depth of the object below the surface of water? What is the index of refraction for water implied by that relationship?

- There exists a mathematical relationship between the distance of an object in front of a mirror and the distance of the image "behind" the mirror.
- There exists a mathematical relationship between the actual depth of an object and the apparent depth of an object seen below the surface of the water.

The word "image" is related to the word "imaginary." The image of the object is "not really there."

Work in teams of three to four students - you will need the extra hands during the refraction portion of the laboratory.

Place an object in front of the mirror. Measure the apparent depth `i` of the image of object "behind" the mirror. Measure the distance `o` from the object to the mirror. Record the distances in centimeters.

Image depth i (cm) | Object depth o (cm) |
---|---|

When an object is underwater, the object appears closer. The actual depth of the object is called the **object depth o** in this laboratory. Measure all distances from the top surface of the water or the glass. The object, especially when viewed from an angle above the water, will appear to be shallower that the actual depth

Measure the object depth `o` and the image depth `i` for the object under the water. Start with an empty tub, recording both the object depth `o` and the image depth `i`. Add water and repeat the measurements. Continue until the tub is full. Record the distances in centimeters.

Make measurements at many different depths. Run an mathematical analysis on the results, producing the appropriate graphs and mathematical relations. Note that the image depth `i` will be the `x` variable while the object depth `o` will be the y variable.

Image depth i (cm) | Object depth o (cm) |
---|---|

Make two xy scattergraphs using spreadsheet software, one for the reflection data and one for the refraction data.

If the graphs indicate that a linear relationship exists for either reflection and/or refraction, find the slopes and intercepts. Use either the LINEST function or, if you know how and wish to do so, use the SLOPE and INTERCEPT functions. If one or both of the graphs suggest either a non-linear or no relationship, note that as well. If there is a relationship, either linear or non-linear, then a mathematical relationship does exist.

=LINEST(y-values;x-values;0)

For the mirror, the theory suggests that the object distance `o` should be equal to the image distance `i`. In theory, the slope should be close to one if this is true. Is your slope close to one? Does your data support this theory? Calculate the difference from a slope of one using the percentage difference formula,
$\text{percentage difference}=\frac{(\text{slope}-1)}{1}$

For the water, the theory suggests that the object distance `o` should be greater than the image distance `i`. In theory, the relation between the image depth `i` and the object depth `o` should obey the equation:

[object deptho] = 1.33 × [image depthi]

Is your slope close to 1.33? Does your data support this theory? Calculate the difference from a slope of one using the percentage difference formula, $\text{percentage difference}=\frac{(\text{slope}-1.33)}{1.33}$

Discuss the findings. Is there a mathematical relationship? Is the relationship linear or non-linear? Discuss whether the hypotheses confirmed or disconfirmed. Discuss also any difficulties you encountered. Are your slopes reasonably close to (percentage differcence ≤ 10%) to the theoretically predicted values? Does your data support or refute those values?

The lab will also be marked on grammar [G], vocabulary [V], organization [O], and cohesion [C].