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Scientific Methods Worksheet 2:
Proportional Reasoning
Some problems adapted from Gibbs' Qualitative Problems for Introductory Physics
1. One hundred cm are equivalent to 1 m. How many cm are equivalent to 3 m? Briefly explain how you could convert any number of meters into a number of centimeters.
3m = 300cm One could set up a proportion as in EMBED Equation.DSMT4 , and solve for x, but it is simpler (and mathematically equivalent) to multiply the number of meters by the factor: EMBED Equation.DSMT4 , as in EMBED Equation.DSMT4
2. Forty-five cm are equivalent to how many m? Briefly explain how you could convert any number of cm into a number of m.
45 cm = 0.45 m Again, one could use a proportion EMBED Equation.DSMT4 and solve for x, but that is a 2-step process wheras one could simply make the conversion by multiplying the number of centimeters by the factor EMBED Equation.DSMT4 , as in EMBED Equation.DSMT4
3. One mole of water is equivalent to 18 grams of water. A glass of water has a mass of 200 g. How many moles of water is this? Briefly explain your reasoning.
EMBED Equation.DSMT4 Multiplying by this factor cancels out the g, leaving moles as the new unit.
Use the metric prefixes table to answer the following questions:
4. The radius of the earth is 6378 km. What is the diameter of the earth in meters?
EMBED Equation.DSMT4
5. In an experiment, you find the mass of a cart to be 250 grams. What is the mass of the cart in kilograms?
EMBED Equation.DSMT4
6. How many megabytes of data can a 4.7 gigabyte DVD store?
Rather than try to figure out how many megabytes are in a gigabytes (or vice-versa), convert to the base unit with one factor and to the desired unit with a second factor. (use B for byte)
EMBED Equation.DSMT4
7. A mile is farther than a kilometer. Consider a fixed distance, like the diameter of the moon. Would the number expressing this distance be larger in miles or in kilometers? Explain.
One would need more of a smaller unit than of a larger unit for a fixed distance. Consider an easier example. A yard is 3 ft but 36 in.
8. One US dollar = 0.73 Euros (as of 12/13.) Which is worth more, one dollar or one Euro? How many dollars is one Euro?
Since one dollar gets you less than one Euro, the Euro is worth more.
EMBED Equation.DSMT4
9. In 2012, Germans paid 1.65 Euros per liter of gasoline. At the same time, American prices were $3.90 per gallon.
a. How much would one gallon of European gas have cost in dollars?
b. How much would one liter of American gasoline have cost in Euros?
(One US dollar = 0.73 Euros, 1 gallon = 3.78 liters)
EMBED Equation.DSMT4
EMBED Equation.DSMT4
10. A mile is equivalent to 1.6 km. When you are driving at 60 miles per hour, what is your speed in meters per second? Clearly show how you used proportions to arrive at a solution.
EMBED Equation.DSMT4 This approach uses unitary factors.
These factors simplify to EMBED Equation.DSMT4 , which is a useful factor to keep in mind for converting speed in miles/hr to m/s.
11. For each of the following mathematical relations, state what happens to the value of y when the following changes are made. (k is a constant)
a. y = kx , x is tripled.
If k is a constant, then the relationship becomes y ( x. Tripling x also triples y.
b. y = k/x, x is halved
If k is a constant, then the relationship becomes y ( 1/x. Halving x doubles y.
EMBED Equation.DSMT4 . The (?) must be replaced by 2.
c. y = k/x2 , x is doubled
If k is a constant, then the relationship becomes y ( 1/x2. Doubling x decreases y by a factor of 4.
EMBED Equation.DSMT4 . The (?) must be replaced by 1/4.
d. y = kx2, x is tripled.
If k is a constant, then the relationship becomes y ( x2. Tripling x increases y by a factor of 9.
EMBED Equation.DSMT4 The (?) must be replaced by a 9.
12. When one variable is directly proportional to another, doubling one variable also doubles the other. If y and x are the variables and a and b are constants, circle the following relationships that are direct proportions. For those that are not direct proportions, explain what kind of proportion does exist between x and y.
a. y = 3x
b. y = ax + b
c. y = x
d. y = ax2
e. y = a/x
f. y = ax
g. y = 1/x
h. y = a/x2
13. The diagram shows a number of relationships between x and y.
a. Which relationships are linear? Explain.
(b), (d), (e) and (f) are linear relationships
The slope of these graphs are constant.
b. Which relationships are direct proportions? Explain.
(b) and (e) - The lines pass through the origin, so a change in x produces the same size change in y.
c. Which relationships are inverse proportions? Explain.
(a) As x increases, y decreases by a constant factor.
Modeling Instruction - AMTA 2013 PAGE 1 U2 Scientific - ws 2 v3.1
Metric prefixes:
giga = 1 000 000 000 billion
mega =1 000 000 million
kilo =1 000 thousand
centi = 1 / 100 hundredth
milli =1 / 1000 thousandth
micro =1 / 1 000 000 millionth
nano =1 / 1 000 000 000 billionth
x
(a), (c), (f) are direct proportions. In (b), y and x are linearly related, but not proportional.
In (d), y is proportional to the square of x.
In (e) and (g), y is inversely proportional to x.
In (h) y is inversely proportional to the square of x.
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Red 1 m 1 0 0 c m E q u a t i o n N a t i v e + _ 1 3 2 2 3 0 9 5 2 5 F 9ۙۙ O l e
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Red 4 5 c m O b j I n f o ! 3 E q u a t i o n N a t i v e 4 Y _ 1 3 2 2 3 0 9 7 0 1 $ F (ۙfۙ O l e
: 1 m 1 0 0 c m = = 0 . 4 5 m F Equation DNQE Equation.DSMT4 ` DSMT6 d MacRoman Arial Symbol Courier Times MT Extra !/ED/APG_AP !APA C o m p O b j # % ; H O b j I n f o & = E q u a t i o n N a t i v e > | _ 1 3 2 2 3 1 0 5 6 5 , ) F ۙۙ E%B_A C_A E*_HA @AHA*_D_E_E_ABBB
Red 2 0 0 g 1 m o l e 1 8 g = = 1 1 . 1 m o l e s F Equation DNQE Equation.DSMT4 O l e
D C o m p O b j ( * E H O b j I n f o + G E q u a t i o n N a t i v e H DSMT6 d MacRoman Arial Symbol Courier Times MT Extra !/ED/APG_AP !APA E%B_A C_A E*_HA @AHA*_D_E_E_ABBB
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p C o m p O b j < > q H O b j I n f o ? s E q u a t i o n N a t i v e t F Equation DNQE Equation.DSMT4 DSMT6 d MacRoman Arial Symbol Courier Times MT Extra !/ED/APG_AP !APA E%B_A C_A E*_HA @AHA*_D_E_E_ABBB
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