Goals
Goals for today: Look at two different ways of setting up unit conversion problem:
- Setting up a proportionality equation
- Using units to think about unit conversion factors
...and practicing these techniques.
Returning to measuring the size of someone in a picture
In the first lab, you'll need to measure a distance that someone walks on one frame of a video image, and see how long it takes them.
On one frame of the image, we can pick out two points, that they'll walk past, let's call them $A$ and $B$, and then we can measure the on-screen distance between them.
I choose points on the sidewalk right below the centers of two trees.
and I found the distance between $A$ and $B$ was $\Delta d=9.6$ "screen" cm (or 96 mm).
There is no yard stick in this picture. But we need a reference object. But, if we know the length of *some* object on the screen, maybe we can convert screen centimeters to real-life centimeters (or inches, or whatever).
I measured the woman's height on screen to be $\Delta h=2.0$ cm. I estimated that her height in real life was pretty close to 5 ft 10 inches=(5*12)+10=70 inches. So, now we have the ratio between real life distances and on screen distances. It is 70 inches (real life) for every 2.0 cm on-screen. Expressing this as a fraction:
$$\frac{\text{70 inches}}{\text{2.0 video cm}}= \frac{\text{35 in}}{\text{1 screen cm}}.$$
There are two ways to think about finding the distance she moved to real-life distances:
1. Proportional reasoning
Let's call the distance moved from A to B in real life "$x$". Since we have a conversion factor to get from screen cm to real life inches, we'll say $x$ has units of inches as well.
The ratio of real-life to screen lengths for the A-B distance should be the same as the ratio of real-life to screen lengths for the woman's height. [As long as....]: $$\frac{x \text{ in}}{9.6\text{ screen cm}}=\frac{\text{70 in}}{\text{2 screen cm}}.$$
This equation can also be stated as "$x$ is to 70 inches as 9.5 cm is to 2.0 cm".
Now we can solve this for $x$, the unknown distance in real life in inches. Multiplying both sides of the equation by 9.6 cm, we get: $$x=9.6*\frac{70}{2}=336$$ Since we determined that $x$ was in inches, the answer is 336 inches.
2. Conversion factor reasoning
We want to convert the on-screen A-B distance, 9.6 screen cm. I'd like to see what that's equal to in real life inches. So I write an algebraic "conversion equation" like this to start with screen cm, multiply by a conversion factor, and end up with an answer in inches: $$9.6\text{ screen cm} *\frac{\text{___inches}}{\text{___screen cm}} = \text{ ____ inches in real life}.$$ Make sure that you write a conversion factor with the right units on top and bottom, so that algebraically, the units on the left and right sides of the equation will be equal.
Our conversion factor is 70 inches for every 2 cm on-screen, so I fill in the appropriate numbers by their units in my equation: $$9.6\text{ screen cm} *\color{red}{\frac{\text{70 inches}}{\text{2.0 screen cm}}} = \text{ ____ inches in real life}.$$
Finally, I multiply and divide through be those numbers and find: $$9.6 * \frac{70}{2} = 336 \text{ inches}.$$
Note that both methods arrived at the same answer!
Practice
Let's do some conversions related to the 100 meter dash. I'd like to know: "About how fast in miles/hour is a really good 100 m dash runner going?". But we'll break this question down into pieces.
Do these calculations on a separate piece of paper, and show all your work. Use either proportional reasoning or conversion factor reasoning.
1.) There are 1000 m in 1 km
Which is larger, a meter or a kilometer?
100 m = < how many > km?
Check your answer by asking... which is larger? a meter or a kilometer? So, my number of kilometers should be less than or more than 100?
[Look again at your conversion factor: the number of kilometers (1) is smaller than the number of meters (1000). So, your answer in kilometers should be less than your number of meters, 100. If not, go back and re-examine what you did.]
2.) 1 km is approximately 0.62 miles
So, < your number of km from > = < how many > miles?
Check your answer: Your answer in miles should more than or less than the number of km's you started with?
3.) Seconds to hours
What's a good time for a human to run a 100 m race?
Convert that number of seconds to *minutes*. Check your answer.
Convert that number of minutes to *hours*. Check your answer.
4.) Put it all together
How fast (in miles per hour) is someone going who is running a good 100 m race?: Take your 100 m, converted to miles and divide by your time (in sec) converted to hours.