Whiteboard: Units are your Friend! The Vietnamese iguana clams at a rate of 8 crankers per week. How many crankers does the iguana clam in a span of 4 months? Hint: You don’t need to know any physics to figure this out
8 crankers week What units do we want our answer to have? crankers What units do our given quantities have? crankers/week weeks 4 weeks = 32 crankers
By using units to your advantage, you can: Check your work Make an educated guess if you’re stuck Help yourself remember an equation that you have forgotten If the units of your result don’t work out to what you wanted, you need to go back and check your work!
Converting units of area and volume A standard sheet of paper is 8.5 inches wide and 11 inches long. How many square feet are in a sheet of paper? 8.5 in x 11 in = 93.5 in 2 How many square inches are in a square foot?
When working with units of area, you must square the conversion factor for length 1 ft = 12 in 1 ft 2 = 12 2 in 2 = 144 in 2 93.5 in 2 = 0.65 ft 2 1 ft 2 144 in 2
Your Turn! How many cubic millimeters are in a cubic meter? Produce two different (yet equivalent) ways of producing the conversion factor!
Answer: 1 x 10 9 mm 3 When working with units of volume, you need to cube the length conversion factors Method 1: 1 m = 1,000 mm 1 m 3 = (1,000 mm) * (1,000 mm) * (1,000 mm) 1 m 3 = 1 x 10 9 mm 3 Method 2: 1 m = 1,000 mm (1 m) 3 = (1,000 mm) 3 1 m 3 = 1,000 3 mm 3 1 m 3 = 1 x 10 9 mm 3
Proportional Reasoning: “Making difficult problems into easy ones!” ® A car that is traveling at 20 mi/hr requires at least 2 feet to stop at full braking force. If the car were instead traveling at 100 mi/hr with all other factors kept the same, in how many feet could it stop? BUT, with a little proportional reasoning, this problem is quick and easy!
v f 2 = v 0 2 + 2a x Car comes to a stop: v f = 0 m/s 0 = v 0 2 + 2a x x = -v 0 2 /(2a) This means that the car’s stopping distance ( x) is proportional to the square of its initial velocity (v 0 2 ) x α v 0 2
This means that whatever happens to v 0, the square of that will happen to x! 20 mi/hr 100 mi/hr x5 2 ft 50 ft x25
Try it – you’ll like it! Two cars on a racetrack each start off at rest at the origin and have identical accelerations. The only difference between them is that car B accelerates for four times as long as car A. If car A’s position at the end of accelerating is 20 m, what is car B’s position after it is finished accelerating? You will need to reason using the equation x f = x o + v 0 t + ½at 2
“start out at rest at the origin”: x 0 = 0, v 0 = 0 x f =½at 2 x f α t 2 If car B is accelerating for four times as long, it will have gone sixteen times as far as car A This means that whatever happens to t, the square of that will happen to x f !
Car A: x f = 20 m Car B: x f = 20 x 16 = 320 m If you are ever asked to compare two objects, or you initially feel that you don’t have enough information to solve a problem, chances are that you will need to use proportional reasoning to solve it! Most students are never taught how to use it – the world is yours! You can use it throughout your college education to save time and impress your professors At first, you may want to crunch the numbers to check your answer as you become comfortable with using proportional reasoning.